T H I R D E D I T I O N
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
JOHN WILEY & SONS, INC. New York Chichester Brisbane Toronto Singapore
T H I R D E D I T I O N
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
JOHN WILEY & SONS, INC. New York Chichester Brisbane Toronto Singapore
ACQUISITIONS EDITOR Cliff Robichaud
ASSISTANT EDITOR Catherine Beckham
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Library of Congress CataloginginPublication Data
Etkin, Bernard. Dynamics of flight : stability and control 1 Bernard Etkin, Lloyd
Duff Reid.3rd ed. p. cm.
Includes bibliographical references (p. ). ISBN 047 10341 85 (cloth : alk. paper) 1. Aerodynamics. 2. Stability of airplanes. I. Reid, Lloyd D.
11. Title.
9520395 CIP
Printed in the United States of America
To the men and women of science and engineering whose contributions to aviation have made it a dominant force in shaping the destiny of mankind, and who, with sensitiviv
and concern, develop and apply their technological arts toward bettering the future.
P R E F A C E
The first edition of this book appeared in 1959indeed before most students reading this were born. It was well received both by students and practicing aeronautical en gineers of that era. The pace of development in aerospace engineering during the decade that followed was extremely rapid, and this was reflected in the subject of flight mechanics. The first author therefore saw the need at the time for a more ad vanced treatment of the subject that included the reality of the round rotating Earth and the real unsteady atmosphere, and hypersonic flight, and that reflected the explo sive growth in computing power that was then taking place (and has not yet ended!). The result was the 1972 volume entitled Dynamics of Atmospheric Flight. That treat ment made no concessions to the needs of undergraduate students, but attempted rather to portray the state of the art of flight mechanics as it was then. To meet the needs of students, a second edition of the 1959 book was later published in 1982. It is that volume that we have revised in the present edition, although in a number of de tails we have preferred the 1972 treatment, and used it instead.
We have retained the same philosophy as in the two preceding editions. That is, we have emphasized basic principles rooted in the physics offlight, essential analyti cal techniques, and typical stability and control realities. We continue to believe, as stated in the preface to the 1959 edition, that this is the preparation that students need to become aeronautical engineers who can face new and challenging situations with confidence.
This edition improves on its predecessors in several ways. It uses a real jet trans port (the Boeing 747) for many numerical examples and includes exercises for stu dents to work in most chapters. We learned from a survey of teachers of this subject that the latter was a sine qua non. Working out these exercises is an important part of acquiring skill in the subject. Moreover, some details in the theoretical development have been moved to the exercises, and it is good practice in analysis for the students to do these.
Students taking a course in this subject are assumed to have a good background in mathematics, mechanics, and aerodynamics, typical of a modem university course in aeronautical or aerospace engineering. Consequently, most of this basic material has been moved to appendices so as not to interrupt the flow of the text.
The content of Chapters 1 through 3 is very similar to that of the previous edi tion. Chapter 4, however, dealing with the equations of motion, contains two very significant changes. We have not presented the nondimensional equations of motion, but have left them in dimensional form to conform with current practice, and we have expressed the equations in the state vector form now commonly used. Chapter 5, on stability derivatives, is almost unchanged from the second edition, and Chapters 6 and 7 dealing with stability and open loop response, respectively, differ from their predecessors mainly in the use of the B747 as example and in the use of the dimen sional equations. Chapter 8, on the other hand, on closed loop control, is very much expanded and almost entirely new. This is consistent with the much enhanced impor tance of automatic flight control systems in modern airplanes. We believe that the
vii
viii Preface
student who works through this chapter and does the exercises will have a good grasp of the basics of this subject.
The appendices of aerodynamic data have been retained as useful material for teachers and students. The same caveats apply as formerly. The data are not intended for design, but only to illustrate orders of magnitude and trends. They are provided to give students and teachers ready access to some data to use in problems and projects.
We acknowledge with thanks the assistance of our colleague, Dr. J. H. de Leeuw, who reviewed the manuscript of Chapter 8 and made a number of helpful sugges tions.
On a personal noteas the first author is now in the 1 lth year of his retirement, this work would not have been undertaken had Lloyd Reid not agreed to collaborate in the task, and if Maya Etkin had not encouraged her husband to take it on and sup ported him in carrying it out.
In turn, the second author, having used the 1959 edition as a student (with the first author as supervisor), the 1972 text as a researcher, and the 1982 text as a
. teacher, wa,s both pleased and honored to work with Bernard Etkin in producing this most recent Gtrsion of the book.
Toronta ; \ Bernard Etkin December, 1994 Lloyd Duff Reid
>
C O N T E N T S
CHAPTER i Introduction
1.1 The Subject Matter of Dynamics of Flight 1 1.2 The Tools of Flight Dynamicists 5
1.3 Stability, Control, and Equilibrium 6 1.4 The Human Pilot 8
1.5 Handling Qualities Requirements 1 1 1.6 Axes and Notation 15
CHAPTER 2 Static Stability and ControlPart 1
General Remarks 18 Synthesis of Lift and Pitching Moment 23 Total Pitching Moment and Neutral Point 29 Longitudinal Control 33 The Control Hinge Moment 41 Influence of a Free Elevator on Lift and Moment 44 The Use of Tabs 47
Control Force to Trim 48
Control Force Gradient 5 1
Exercises 52
Additional Symbols Introduced in Chapter 2 57
CHAPTER 3 Static Stability and ControlPart 2
3.1 ManeuverabilityElevator Angle per g 60 3.2 Control Force per g 63 3.3 Influence of HighLift Devices on Trim and Pitch Stiffness 64 3.4 Influence of the Propulsive System on Trim and Pitch Stiffness 66 3.5 Effect of Structural Flexibility 72
3.6 Ground Effect 74 3.7 CG Limits 74
3.8 Lateral Aerodynamics 76 3.9 Weathercock Stability (Yaw Stiffness) 77 3.10 Yaw Control 80 3.11 Roll Stiffness 81 3.12 The Derivative C,, 83 3.13 Roll Control 86 3.14 Exercises 89 3.15 Additional Symbols Introduced in Chapter 3 9 1
CHAPTER 4 General Equations of Unsteady Motion
4.1 General Remarks 93 4.2 The RigidBody Equations 93
x Contents
Evaluation of the Angular Momentum h 96 Orientation and Position of the Airplane 98 Euler's Equations of Motion 100 Effect of Spinning Rotors on the Euler Equations 103 The Equations Collected 103 Discussion of the Equations 104 The SmallDisturbance Theory 107 The Nondimensional System 115 Dimensional Stability Derivatives 1 18 Elastic Degrees of Freedom 120 Exercises 126 Additional Symbols Introduced in Chapter 4 127
CHAPTER 5 The Stability Derivatives
General Remarks 129 The a Derivatives 129 The u Derivatives 131 The q Derivatives 135 The & Derivatives 141 The P Derivatives 148 The p Derivatives 149 The r Derivatives 153 Summary of the Formulas 154 Aeroelastic Derivatives 156 Exercises 159 Additional Symbols Introduced in Chapter 5 160
CHAPTER 6 Stability of Uncontrolled Motion
Form of Solution of SmallDisturbance Equations 161 Longitudinal Modes of a Jet Transport 165 Approximate Equations for the Longitudinal Modes 171 General Theory of Static Longitudinal Stability 175 Effect of Flight Condition on the Longitudinal Modes of a Subsonic Jet Transport 177 Longitudinal Characteristics of a STOL Airplane 184 Lateral Modes of a Jet Transport 187 Approximate Equations for the Lateral Modes 193 Effects of Wind 196
Exercises 201 Additional Symbols Introduced in Chapter 6 203
CHAPTER 7 Response to Actuation of the ControlsOpen Loop
7.1 General Remarks 204 7.2 Response of LinearIInvariant Systems 207 7.3 Impulse Response 210
Contents xi
StepFunction Response 21 3 Frequency Response 2 14 Longitudinal Response 228 Responses to Elevator and Throttle 229 Lateral Steady States 237 Lateral Frequency Response 243 Approximate Lateral Transfer Functions 247 Transient Response to Aileron and Rudder 252 Inertial Coupling in Rapid Maneuvers 256 Exercises 256 Additional Symbols Introduced in Chapter 7 258
CHAPTER 8 ClosedLoop Control
General Remarks 259 Stability of Closed Loop Systems 264 Phugoid Suppression: Pitch Attitude Controller 266 Speed Controller 270 Altitude and Glide Path Control 275 Lateral Control 280 Yaw Damper 287 Roll Controller 290 Gust Alleviation 295 Exercises 300 Additional Symbols Introduced in Chapter 8 301
APPENDIX A Analytical Tools
A. 1 Linear Algebra 303 A.2 The Laplace Transform 304 A.3 The Convolution Integral 309 A.4 Coordinate Transformations 3 10 A.5 Computation of Eigenvalues and Eigenvectors 3 15 A.6 Velocity and Acceleration in an Arbitrarily Moving Frame 3 16
APPENDIX B Data for Estimating Aerodynamic Derivatives 319
APPENDIX c Mean Aerodynamic Chord, Mean Aerodynamic Center, and c m a c w 357
APPENDIX D The Standard Atmosphere and Other Data 364
APPENDIX E Data For the Boeing 7471 00 369
References 372
Index 377
C H A P T E R 1
Introduction
1.1 The Subject Matter of Dynamics of Flight
This book is about the motion of vehicles that fly in the atmosphere. As such it be longs to the branch of engineering science called applied mechanics. The three itali cized words above warrant further discussion. To begin withflythe dictionary defi nition is not very restrictive, although it implies motion through the air, the earliest application being of course to birds. However, we also say "a stone flies" or "an ar row flies," so the notion of sustention (lift) is not necessarily implied. Even the at mospheric medium is lost in "the flight of angels." We propose as a logical scientific definition that flying be defined as motion through a fluid medium or empty space. Thus a satellite "flies" through space and a submarine "flies" through the water. Note that a dirigible in the air and a submarine in the water are the same from a mechani cal standpointthe weight in each instance is balanced by buoyancy. They are sim ply separated by three orders of magnitude in density. By vehicle is meant any flying object that is made up of an arbitrary system of deformable bodies that are somehow joined together. To illustrate with some examples: (1) A rifle bullet is the simplest kind, which can be thought of as a single ideally rigid body. (2) A jet transport is a more complicated vehicle, comprising a main elastic body (the airframe and all the parts attached to it), rotating subsystems (the jet engines), articulated subsystems (the aerodynamic controls) and fluid subsystems (fuel in tanks). (3) An astronaut attached to his orbiting spacecraft by a long flexible cable is a further complex example of this general kind of system. Note that by the above definition a vehicle does not necessar ily have to carry goods or passengers, although it usually does. The logic of the defi nitions is simply that the underlying engineering science is common to all these ex amples, and the methods of formulating and solving problems concerning the motion are fundamentally the same.
As is usual with definitions, we can find examples that don't fit very well. There are special cases of motion at an interface which we may or may not include in fly ingfor example, ships, hydrofoil craft and aircushion vehicles (ACV's). In this connection it is worth noting that developments of hydrofoils and ACV's are fre quently associated with the Aerospace industry. The main difference between these cases, and those of "true" flight, is that the latter is essentially threedimensional, whereas the interface vehicles mentioned (as well as cars, trains, etc.) move approxi mately in a twodimensional field. The underlying principles and methods are still the same however, with certain modifications in detail being needed to treat these "surface" vehicles.
Now having defined vehicles andflying, we go on to look more carefully at what we mean by motion. It is convenient to subdivide it into several parts:
Aerodynamics
Mechanics of Veh~cle r~gid bodies design
Mechan~cs of FLIGHT Vehlcle elastic structures DYNAMICS operation
Human p~ lo t Pilot dynamics t ra~n~ny
Applied mathematics. machlne computatlon
3. 4 4  Performance Stablllty and Aeroelasticity (trajectory. control (handl~ny (control, structural
Navigation and
maneuverability) qual~ties, a~rloads) integrity) guidance
 Figure 1.1 Block diagram of disciplines.
Gross Motion:
1 . Trajectory of the vehicle mass center.'
2. "Attitude" motion, or rotations of the vehicle "as a whole."
Fine Motion:
3. Relative motion of rotating or articulated subsystems, such as engines, 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 sociated with the different motions, and of the formulation of their analysis. It is surely selfevident that studies of these motions must be central to the design and op eration of aircraft, spacecraft, rockets, missiles, etc. To be able to formulate and solve the relevant problems, we must draw on several basic disciplines from engineering science. The relationships are shown on Fig. 1 .l. It is quite evident from this figure that the practicing flight dynamicist requires intensive training in several branches of engineering science, and a broad outlook insofar as the practical ramifications of his work are concerned.
In the classes of vehicles, in the types of motions, and in the medium of flight, this book treats a very restricted set of all possible cases. It deals only with the flight
'It is assumed that gravity is uniform, and hence that the mass center and center of gravity (CG) are the same point.
1.1 The Subject Matter of Dynamics of Flight 3
of airplanes in the atmosphere. The general equations derived, and the methods of so lution presented, are however readily modified and extended to treat many of the other situations that are embraced by the general problem.
All the fundamental science and mathematics needed to develop this subject ex isted in the literature by the time the Wright brothers flew. Newton, and other giants of the 18th and 19th centuries, such as Bernoulli, Euler, Lagrange, and Laplace, pro vided the building blocks in solid mechanics, fluid mechanics, and mathematics. The needed applications to aeronautics were made mostly after 1900 by workers in many countries, of whom special reference should be made to the Wright brothers, G. H. Bryan, F. W. Lanchester, J. C. Hunsaker, H. B. Glauert, B. M. Jones, and S. B. Gates. These pioneers introduced and extended the basis for analysis and experiment that underlies all modern p ra~ t i ce .~ This body of knowledge is well documented in several texts of that period, for example, Bairstow (1939). Concurrently, principally in the United States of America and Britain, a large body of aerodynamic data was accumu lated, serving as a basis for practical design.
Newton's laws of motion provide the connection between environmental forces and resulting motion for all but relativistic and quantumdynamical processes, includ ing all of "ordinary" and much of celestial mechanics. What then distinguishes flight dynamics from other branches of applied mechanics? Primarily it is the special na ture of the force fields with 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 fields may be identified as follows:
"Strong" Fields:
1. Gravity
2. Aerodynamic
3. Buoyancy
"Weak" Fields:
4. Magnetic
5. Solar radiation
We should observe that two of these fields, aerodynamic and solar radiation, pro duce important heat transfer to the vehicle in addition to momentum transfer (force). Sometimes we cannot separate the thermal and mechanical problems (Etkin and Hughes, 1967). Of these fields only the strong ones are of interest for atmospheric and oceanic flight, the weak fields being important only in space. It should be re marked that even in atmospheric flight the gravity force can not always be approxi mated as a constant vector in an inertial frame. Rotations associated with Earth cur vature, and the inverse square law, become important in certain cases of highspeed and highaltitude flight (Etkin, 1972).
The prediction, measurement and representation of aerodynamic forces are the principal distinguishing features of flight dynamics. The size of this task is illustrated
2An excellent account of the early history is given in the 1970 von Kirmin Lecture by Perkins (1970).
4 Chapter I . Introduction
Parameters of wing aerodynamics
SHAPE: sections Wings  0 0 a 0 1 .O 5.0
SPEED: I I *M Subsonic Supersonic Hypersonic
Incompressible Transonic
MOTION: Constant velocity Variable velocity
[u, v, w, p, q, r] = const Iu(t), v(t), w(t), p(t), r(t)l
ATMOSPHERE: I I I Continuum Slip Freemolecule
Uniform and Nonuniform and Uniform and onu uniform and at rest at rest in motion in motion
(reentry) (gusts)
Figure 1.2 Spectrum of aerodynamic problems for wings.
by Fig. 1.2, which shows the enormous range of variables that need to be considered in connection with wings alone. To be added, of course, are the complications of propulsion systems (propellers, jets, rockets), compound geometries (wing + body + tail), and variable geometry (wing sweep, camber).
As remarked above, Newton's laws state the connection between force and mo tion. The commonest problem consists of finding the motion when the laws for the forces are given (all the numerical examples given in this book are of this kind). However, we must be aware of certain important variations:
1. Inverse problems of first kindthe system and the motion are given and the forces have to be calculated.
2. Inverse problems of the second kindthe forces and the motion are given and the system parameters have to be found.
3. Mixed problemsthe unknowns are a mixture of variables from the force, system, and motion.
Examples of these inverse and mixed problems often turn up in research, when one is trying to deduce aerodynamic forces from the observed motion of a vehicle in flight or of a model in a wind tunnel. Another example is the deduction of harmonics of the Earth's gravity field from observed perturbations of satellite orbits. These problems are closely related to the "plant identification" or "parameter identification" problem of system theory. [Inverse problems were treated in Chap. 11 of Etkin (1959)l.
1.2 The Tools of Flight Dynarnicists 5
TYPES OF PROBLEMS
The main types of flight dynamics problem that occur in engineering practice are:
1. Calculation of "performance" quantities, such as speed, height, range, and fuel consumption.
2. Calculation of trajectories, such as launch, reentry, orbital and landing.
3. Stability of motion.
4. Response of vehicle to control actuation and to propulsive changes.
5 . Response to atmospheric turbulence, and how to control it.
6. Aeroelastic oscillations (flutter).
7. Assessment of humanpilotlmachine combination (handling qualities).
It takes little imagination to appreciate that, in view of the many vehicle types that have to be dealt with, a number of subspecialties exist within the ranks of flight dynamicists, related to some extent to the above problem categories. In the context of the modern aerospace industry these problems are seldom simple or routine. On the contrary they present great challenges in analysis, computation, and experiment.
1.2 The Tools of Flight Dynamicists
The tools used by flight dynamicists to solve the design and operational problems of vehicles are of three kinds:
1. Analytical
2. Computational
3. Experimental
The analytical tools are essentially the same as those used in other branches of mechanics, that is the methods of applied mathematics. One important branch of ap plied mathematics is what is now known as system theory, including stability, auto matic control, stochastic processes and optimization. Stability of the uncontrolled ve hicle is neither a necessary nor a sufficient condition for successful controlled flight. Good airplanes have had slightly unstable modes in some part of their flight regime, and on the other hand, a completely stable vehicle may have quite unacceptable han dling qualities. It is dynamic peijormance criteria that really matter, so to expend a great deal of analytical and computational effort on finding stability boundaries of nonlinear and timevarying systems may not be really worthwhile. On the other hand, the computation of stability of small disturbances from a steady state, that is, the lin ear eigenvalue problem that is normally part of the system study, is very useful in deed, and may well provide enough information about stability from a practical standpoint.
On the computation side, the most important fact is that the availability of ma chine computation has revolutionized practice in this subject over the past few decades. Problems of system performance, system design, and optimization that
6 Chapter I . Introduction
could not have been tackled at all in the past are now handled on a more or less rou tine basis.
The experimental tools of the flight dynamicist are generally unique to this field. First, there are those that are used to find the aerodynamic inputs. Wind tunnels and shock tubes that cover most of the spectrum of atmospheric flight are now available in the major aerodynamic laboratories of the world. In addition to fixed laboratory equipment, there are aeroballistic ranges for dynamic investigations, as well as rocketboosted and gunlaunched freeflight model techniques. Hand in hand with the development of these general facilities has gone that of a myriad of sensors and in struments, mainly electronic, for measuring forces, pressures, temperatures, accelera tion, angular velocity, and so forth. The evolution of computational fluid dynamics (CFD) has sharply reduced the dependence of aerodynamicists on experiment. Many results that were formerly obtained in wind tunnel tests are now routinely provided by CFD analyses. The CFD codes themselves, of course, must be verified by compar ison with experiment.
Second, we must mention the flight simulator as an experimental tool used di rectly by the flight dynamicist. In it he studies mainly the matching of the pilot to the machine. This is an essential step for radically new flight situations. The ability of the pilot to control the vehicle must be assured long before the prototype stage. This can not yet be done without test, although limited progress in this direction is being made through studies of mathematical models of human pilots. Special simulators, built for most new major aircraft types, provide both efficient means for pilot training, and a research tool for studying handling qualities of vehicles and dynamics of human pi lots. The development of highfidelity simulators has made it possible to greatly re duce the time and cost of training pilots to fly new types of airplanes.
1.3 Stability, Control, and Equilibrium
It is appropriate here to define what is meant by the terms stability and control. To do so requires that we begin with the concept of equilibrium.
A body is in equilibrium when it is at rest or in uniform motion (i.e., has constant linear and angular momenta). The most familiar examples of equilibrium are the static ones; that is, bodies at rest. The equilibrium of an airplane in flight, however, is of the second kind; that is, uniform motion. Because the aerodynamic forces are de pendent on the angular orientation of the airplane relative to its flight path, and be cause the resultant of them must exactly balance its weight, the equilibrium state is without rotation; that is, it is a motion of rectilinear translation.
Stability, or the lack of it, is a property of an equilibrium state.3 The equilibrium is stable if, when the body is slightly disturbed in any of its degrees of freedom, it re turns ultimately to its initial state. This is illustrated in Fig. 1.3a. The remaining sketches of Fig. 1.3 show neutral and unstable equilibrium. That in Fig. 1.3d is a more complex kind than that in Fig. 1.36 in that the ball is stable with respect to dis placement in the y direction, but unstable with respect to x displacements. This has its counterpart in the airplane, which may be stable with respect to one degree of free dom and unstable with respect to another. Two kinds of instability are of interest in
'It is also possible to speak of the stability of a transient with prescribed initial condition.
1.3 Stability, Control, and Equilibrium 7
Figure 1.3 (a) Ball in a bowlstable equilibrium. (b) Ball on a hillunstable equilibrium. (c ) Ball on a planeneutral equilibrium. (d) Ball on a saddle surfaceunstable equilibrium.
airplane dynamics. In the first, called static instability, the body departs continuously from its equilibrium condition. That is how the ball in Fig. 1.3b would behave if dis turbed. The second, called dynamic instability, is a more complicated phenomenon in which the body oscillates about its equilibrium condition with everincreasing ampli tude.
When applying the concept of stability to airplanes, there are two classes that must be consideredinherent stability and synthetic stability. The discussion of the previous paragraph implicitly dealt with inherent stability, which is a property of the basic airframe with either fixed or free controls, that is, controlfixed stability or controlfree stability. On the other hand, synthetic stability is that provided by an au tomatic flight control system (AFCS) and vanishes if the control system fails. Such automatic control systems are capable of stabilizing an inherently unstable airplane, or simply improving its stability with what is known as stability augmentation sys tems (SAS). The question of how much to rely on such systems to make an airplane flyable entails a tradeoff among weight, cost, reliability, and safety. If the SAS works most of the time, and if the airplane can be controlled and landed after it has failed, albeit with diminished handling qualities, then poor inherent stability may be acceptable. Current aviation technology shows an increasing acceptance of SAS in all classes of airplanes.
If the airplane is controlled by a human pilot, some mild inherent instability can be tolerated, if it is something the pilot can control, such as a slow divergence. (Un stable bicycles have long been ridden by humans!). On the other hand, there is no
8 Chapter I . Introduction
margin for error when the airplane is under the control of an autopilot, for then the closed loop system must be stable in its response to atmospheric disturbances and to commands that come from a navigation system.
In addition to the role controls play in stabilizing an airplane, there are two oth ers that are important. The first is to fix or to change the equilibrium condition (speed or angle of climb). An adequate control must be powerful enough to produce the whole range of equilibrium states of which the airplane is capable from a perfor mance standpoint. The dynamics of the transition from one equilibrium state to an other are of interest and are closely related to stability. The second function of the control is to produce nonequilibrium, or accelerated motions; that is, maneuvers. These may be steady states in which the forces and accelerations are constant when viewed from a reference frame fixed to the airplane (for example, a steady turn), or they may be transient states. Investigations of the transition from equilibrium to a nonequilibrium steady state, or from one maneuvering steady state to another, form part of the subject matter of airplane control. Very large aerodynamic forces may act on the airplane when it maneuversa knowledge of these forces is required for the proper design of the structure.
RESPONSE TO ATMOSPHERIC TURBULENCE
A topic that belongs in dynamics of flight and that is closely related to stability is the response of the airplane to wind gradients and atmospheric turbulence (Etkin, 1981). This response is important from several points of view. It has a strong bearing on the adequacy of the structure, on the safety of landing and takeoff, on the acceptability of the airplane as a passenger transport, and on its accuracy as a gun or bombing plat form.
1.4 The Human Pilot
Although the analysis and understanding of the dynamics of the airplane as an iso lated unit is extremely important, one must be careful not to forget that for many flight situations it is the response of the total system, made up of the human pilot and the aircraft, that must be considered. It is for this reason that the designers of aircraft should apply the findings of studies into the human factors involved in order to en sure that the completed system is well suited to the pilots who must fly it.
Some of the areas of consideration include:
1. Cockpit environment; the occupants of the vehicle must be provided with oxygen, warmth, light, and so forth, to sustain them comfortably.
2. Instrument displays; instruments must be designed and positioned to provide a useful and unambiguous flow of information to the pilot.
3. Controls and switches; the control forces and control system dynamics must be acceptable to the pilot, and switches must be so positioned and designed as to prevent accidental operation. Tables 1.1 to 1.3 present some pilot data con cerning control forces.
4. Pilot workload; the workload of the pilot can often be reduced through proper planning and the introduction of automatic equipment.
1.4 The Human Pilot 9
Table 1.1 Estimates of the Maximum Rudder Forces that Can Be Exerted for Various Positions of the Rudder Pedal (BuAer, 1954)
Rudder Pedal Position Distance from Back ($Seat Pedal Force
(in) (cm) (lb) ( N )
Back 3 1 .OO 78.74 246 1,094 Neutral 34.75 88.27 424 1,886 Forward 38.50 97.79 334 1,486
Table 1.2 HandOperated Control Forces (From Flight Safety Foundation Human Engineering Bulletin 565H) (see figure in Table 1.3)
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.
10 Chapter I . Introduction
DIRECTION OF MOVEMENT
Vert, ref, line
180°
90'
Inboard
Table 1.3 Rates of Stick Movement in Flight Test Pullups Under Various Loads (BuAer, 1954)
Maximum Stick Average Rate of Stick Time for Full Pullup Load Motion Deflection
(lb) ( N ) (in'..) (cm/s) (s)
1.5 Handling Qualities Requirements 11
The care exercised in considering the human element in the closedloop system made up of pilot and aircraft can determine the success or failure of a given aircraft design to complete its mission in a safe and efficient manner.
Many critical tasks performed by pilots involve them in activities that resemble those of a servo control system. For example, the execution of a landing approach through turbulent air requires the pilot to monitor the aircraft's altitude, position, atti tude, and airspeed and to maintain these variables near their desired values through the actuation of the control system. It has been found in this type of control situation that the pilot can be modeled by a linear control system based either on classical con trol theory or optimal control theory (Etkin, 1972; Kleinman et al., 1970; McRuer and Krendel, 1973).
1.5 Handling Qualities Requirements
As a result of the inability to carry out completely rational design of the pilot machine combination, it is customary for the government agencies responsible for the procurement of military airplanes, or for licensing civil airplanes, to specify com pliance with certain "handling (or flying) qualities requirements" (e.g., ICAO, 199 1; USAF, 1980; USAF, 1990). Handling qualities refers to those qualities or character istics of an aircraft that govern the ease and precision with which a pilot is able to perform the tasks required in support of an aircraft role (Cooper and Harper, 1969).
These requirements have been developed from extensive and continuing flight research. In the final analysis they are based on the opinions of research test pilots, substantiated by careful instrumentation. They vary from country to country and from agency to agency, and, of course, are different for different types of aircraft. They are subject to continuous study and modification in order to keep them abreast of the lat est research and design information. Because of these circumstances, it is not feasible to present a detailed description of such requirements here. The following is intended to show the nature, not the detail, of typical handling qualities req~irements.~ Most of the specific requirements can be classified under one of the following headings.
CONTROL POWER
The term control power is used to describe the efficacy of a control in producing a range of steady equilibrium or maneuvering states. For example, an elevator control, which by taking positions between full up and full down can hold the airplane in equilibrium at all speeds in its speed range, for all configurations5 and CG positions, is a powerful control. On the other hand, a rudder that is not capable at full deflection of maintaining equilibrium of yawing moments in a condition of one engine out and negligible sideslip is not powerful enough. The handling qualities requirements nor mally specify the specific speed ranges that must be achievable with full elevator de
'For a more complete discussion, see AGARD (1959); Stevens and Lewis (1992)
5This word describes the position of movable elements of the airplanefor example, landing con figuration means that landing flaps and undercarriage are down, climb configuration means that landing gear is up, and flaps are at takeoff position, and so forth.
12 Chapter I. Introduction
flection in the various important configurations and the asymmetric power condition that the rudder must balance. They may also contain references to the elevator angles required to achieve positive load factors, as in steady turns and pullup maneuvers (see "elevator angle per g," Sec. 3.1).
CONTROL FORCES
The requirements invariably specify limits on the control forces that must be exerted by the pilot in order to effect specific changes from a given trimmed condition, or to maintain the trim speed following a sudden change in configuration or throttle set ting. They frequently also include requirements on the control forces in pullup ma neuvers (see "control force per g," Sec. 3.1). In the case of light aircraft, the control forces can result directly from mechanical linkages between the aerodynamic control surfaces and the pilot's flight controls. In this case the hinge moments of Sec. 2.5 play a direct role in generating these forces. In heavy aircraft, systems such as partial or total hydraulic boost are used to counteract the aerodynamic hinge moments and a related or independent subsystem is used to create the control forces on the pilot's flight controls.
STATIC STABILITY
The requirement for static longitudinal stability (see Chap. 2) is usually stated in terms of the neutral point. The neutral point, defined more precisely in Sec. 2.3, is a special location of the center of gravity (CG) of the airplane. In a limited sense it is the boundary between stable and unstable CG positions. It is usually required that the relevant neutral point (stick free or stick fixed) shall lie some distance (e.g., 5% of the mean aerodynamic chord) behind the most aft position of the CG. This ensures that the airplane will tend to fly at a constant speed and angle of attack as long as the controls are not moved.
The requirement on static lateral stability is usually mild. It is simply that the spiral mode (see Chap. 6) if divergent shall have a time to double greater than some stated minimum (e.g., 4s).
DYNAMIC STABILITY
The requirement on dynamic stability is typically expressed in terms of the damping and frequency of a natural mode. Thus the USAF (1980) requires the damping and frequency of the lateral oscillation for various flight phases and stability levels to conform to the values in Table 1.4.
STALLING AND SPINNING
Finally, most requirements specify that the airplane's behavior following a stall or in a spin shall not include any dangerous characteristics, and that the controls must re tain enough effectiveness to ensure a safe recovery to normal flight.
1.5 Handling Qualities Requirements 13
'Level, Phase and Class are defined in USAF, 1980.
*Note: The damping coefficient 4; and the undamped natural frequency w,,, are defined in Chap. 6.
Table 1.4' Minimum Dutch Roll Frequency and Damping
RATING OF HANDLING QUALITIES
3
To be able to assess aircraft handling qualities one must have a measuring technique with which any given vehicle's characteristics can be rated. In the early days of avia tion, this was done by soliciting the comments of pilots after they had flown the air craft. However, it was soon found that a communications problem existed with pilots using different adjectives to describe the same flight characteristics. These ambigui 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 as the "CooperHarper Scale," where a numerical rating scale is utilized in conjunction with a set of descriptive phrases. This scale is presented in Fig. 1.4. To apply this rating 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 (110) is merely a shorthand notation for the descriptive phrases and as such no mathemati cal operations can be carried out on them in a rigorous sense. For example, a vehicle configuration rated as 6 should not be thought to be "twice as bad" as one rated at 3. The comments from evaluation pilots are extremely useful and this information will provide the detailed reasons for the choice of a rating.
Other techniques have been applied to the rating of handling qualities. For exam ple, attempts have been made to use the overall system performance as a rating pa rameter. However, due to the pilot's adaptive capability, quite often he can cause the overall system response of a bad vehicle to approach that of a good vehicle, leading to the same performance but vastly differing pilot ratings. Consequently system per formance has not proved to be a good rating parameter. A more promising approach involves the measurement of the pilot's physiological and psychological state. Such methods lead to objective assessments of how the system is influencing the human controller. The measurement of human pilot describing functions is part of this tech nique (Kleinman et al., 1970; McRuer and Krendel, 1973; Reid, 1969).
Research into aircraft handling qualities is aimed in part at ascertaining which vehicle parameters influence pilot acceptance. It is obvious that the number of possi
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1.6 Axes and Notation 15
I I I I I 6.0  Initial response fast, 
oversensitive, light stick forces

 Sluggish, large stick motion and forces
 C m C

1.0  Unacceptable  to maneuver, difficult to trim
0 1 I I I I I 1 0.1 0.5. 1.0 2.0 3.0 4.0
Damping ratio, f Figure 1.5 Longitudinal shortperiod oscillationpilot opinion contours (O'Hara, 1967).
ble combinations of parameters is staggering, and consequently attempts are made to study one particular aspect of the vehicle while maintaining all others in a "satisfac tory" configuration. Thus the task is formulated in a fashion that is amenable to study. The risk involved in this technique is that important interaction effects can be overlooked. For example, it is found that the degree of difficulty a pilot finds in con trolling an aircraft's lateraldirectional mode influences his rating of the longitudinal dynamics. Such facts must be taken into account when interpreting test results. An other possible bias exists in handling qualities results obtained in the past because most of the work has been done in conjunction with fighter aircraft. The findings from such research can often be presented as "isorating" curves such as those shown in Fig. 1.5.
1.6 Axes and Notation
In this book the Earth is regarded as flat and stationary in inertial space. Any coordi nate system, or frame of reference, attached to the Earth is therefore an inertial sys tem, one in which Newton's laws are valid. Clearly we shall need such a reference frame when we come to formulate the equations of motion of a flight vehicle. We de note that frame by F,(O,,x,,y,,z,). Its origin is arbitrarily located to suit the circum stances of the problem, the axis O,z, points vertically downward, and the axis O,x,, which is horizontal, is chosen to point in any convenient direction, for example, North, or along a runway, or in some reference flight direction. It is additionally as sumed that gravity is uniform, and hence that the mass center and center of gravity (CG) are the same point. The location of the CG is given by its Cartesian coordinates relative to F , . Its velocity relative to F, is denoted V" and is frequently termed the groundspeed.
16 Chapter I. Introduction
Figure 1.6 Notation 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 F,, but rather on the velocity relative to the surrounding air mass (the airspeed), which will differ from the groundspeed whenever there is a wind. If we denote the wind ve locity vector relative to FE by W, and that of the CG relative to the air by V then clearly
V E = V + W (1.6,l)
The components of W in frame FE, that is, relative to Earth, are given by
V represents the magnitude of the airspeed (thus retaining the usual aerodynamics meaning of this symbol). For the most part we will have W = 0, making the airspeed the same as the inertial velocity.
A second frame of reference will be needed in the development of the equations of motion. This frame is fixed to the airplane and moves with it, having its origin C at the CG, (see ~ i ~ . 1.6). It is denoted F, and is commonly called body axes. Cxz is the plane of symmetry of the vehicle. The components of the aerodynamic forces and moments that act on the airplane, and of its linear and angular velocities relative to
X
  Projection of V on xz plane 
Trace of x$ plane
Z (a) (6 LI
Figure 1.7 (a ) Definition of a,. (b) View in plane of y and V, definition of P.
1.6 Axes and Notation 17
the air are denoted by the symbols given in the figure. In the notation of Appendix A. 1, this means, for example, that
v, = [U U w]' (1 6 3 )
The vector V does not in general lie in any of the coordinate planes. Its orienta tion is defined by the two angles shown in Fig. 1.7:
W Angle of attack, a, = tan' 
11
U (1 6 4 )
Angle of sideslip, /3 = sinp' 7
With these definitions, the sideslip angle /3 is not dependent on the direction of Cx in the plane of symmetry.
The symbols used throughout the text correspond generally to current usage and are mainly used in a consistent manner.
C H A P T E R 2
Static Stability and Control Part 1
2.1 General Remarks
A general treatment of the stability and control of airplanes requires a study of the dynamics of flight, and this approach is taken in later chapters. Much useful informa tion can be obtained, however, from a more limited view, in which we consider not the motion of the airplane, but only its equilibrium states. This is the approach in what is commonly known as static stability and control analysis.
The unsteady motions of an airplane can frequently be separated for convenience into two parts. One of these consists of the longitudinal or symmetric motions; that is, those in which the wings remain level, and in which the center of gravity moves in a vertical plane. The other consists of the lateral or asymmetric motions; that is, rolling, yawing, and sideslipping, while the angle of attack, the speed, and the angle of elevation of the x axis remains constant.
This separation can be made for both dynamic and static analyses. However, the results of greatest importance for static stability are those associated with the longitu dinal analysis. Thus the principal subject matter of this and the following chapter is static longitudinal stability and control. A brief discussion of the static aspects of di rectional and rolling motions is contained in Secs. 3.9 and 3. l l .
We shall be concerned with two aspects of the equilibrium state. Under the head ing stability we shall consider the pitching moment that acts on the airplane when its angle of attack is changed from the equilibrium value, as by a vertical gust. We focus our attention on whether or not this moment acts in such a sense as to restore 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.
The restriction to angle of attack disturbances when dealing with stability must be noted, since the applicability of the results is thereby limited. When the aerody namic characteristics of an airplane change with speed, owing to compressibility ef fects, structural distortion, or the influence of the propulsive system, then the airplane may be unstable with respect to disturbances in speed. Such instability is not pre dicted by a consideration of angle of attack disturbances only. (See Fig. 1.3d, and identify speed with x, angle of attack with y.) A more general point of view than that adopted in this chapter is required to assess that aspect of airplane stability. Such a viewpoint is taken in Chap. 6. To distinguish between true general static stability and the more limited version represented by C, vs. a, we use the term pitch stifiess for the latter.
2.1 General Remarks 19
Although the major portion of this and the following chapter treats a rigid air 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 is symmetric steady flight. In this condi tion the velocity and force vectors are as illustrated in Fig. 2.1. All the nonzero forces and motion variables are members of 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 varies with V and a , since it is so dependent on the type of propulsion unitrockets, jet, propeller, or turboprop. Two particular idealizations are of interest, however,
1 . T independent of V, that is, constant thrust; an approximation for rockets and pure jets.
2. 7'V independent of V, that is, constant power; an approximation for reciprocat ing engines with constantspeed propellers.
The variation of steadystate lift and drag with a for subsonic and supersonic Mach numbers (M < about 5) are characteristically as shown in Fig. 2.2 for the range of attached flow over the surfaces of the vehicle (McCormick, 1994; Miele, 1962; Schlichting and Truckenbrodt, 1979). Over a useful range of a (below the stall) the coefficients are given accurately enough by
CL = CL,?ff (2.1,l)
CD = CD,,,," + KCL2 (2.12)
The three constants C , , C,m,,n, K are principally functions of the configuration shape, thrust coefficient, and Mach number.
Significant departure from the above idealizations may, of course, be anticipated in some cases. The minimum of C, may occur at a value of a > 0, and the curvature of the C, vs. a relation may be an important consideration for flight at high CL. When the vehicle is a "slender body," for example, a slender delta, or a slim wingless
Zerolift line
CG ' \
w Figure 2.1 Steady symmetric flight.
20 Chapter 2. Static Stability and ControlPart 1
CL CD
/'
/'
a a
Figure 2.2 L ~ f t and drag for subsonic and supersonic speeds.
body, the CL curve may have a characteristic upward curvature even at small a (Flax and Lawrence, 195 1 ) . The upward curvature of CL at small a is inherently present at hypersonic Mach numbers (Truitt, 1959). For the nonlinear cases, a suitable formula tion for CL is (USAF, 1978)
C, = (tCNa sin 2 a + CNa, sin a /sin a/) cos a (2.1,3)
where CNa and CNea are coefficients (independent of a ) that depend on the Mach number and configuration. [Actually CN here is the coefficient of the aerodynamic force component normal to the wing chord, and CNe is the value of CLa at a = 0, as can easily be seen by linearizing (2.1,3) with respect to a.] Equation (2.1,2) for the drag coefficient can serve quite well for flight dynamics applications up to hyper sonic speeds (M > 5) at which theory indicates that the exponent of CL decreases from 2 to 8. Miele (1962) presents in Chap. 6 a very useful and instructive set of typical lift and drag data for a wide range of vehicle types, from subsonic to hyper sonic.
Balance, or Equilibrium
An airplane can continue in steady unaccelerated flight only when the resultant external force and moment about the CG both vanish. In particular, this requires that the pitching moment be zero. This is the condition of longitudinal balance. If the pitching moment were not zero, the airplane would experience a rotational accelera tion component in the direction of the unbalanced moment. Figure 2.3 shows a typi cal graph of the pitchingmoment coefficient about the CG1 versus the angle of attack for an airplane with a fixed elevator (curve a). The angle of attack is measured from the zerolift line of the airplane. The graph is a straight line except near the stall. Since zero C, is required for balance, the airplane can fly only at the angle of attack marked A, for the given elevator angle.
'Unless otherwise specified, C , always refers to moment about the CG.
2.1 General Remarks 21
I Balanced a l d positive stiffness UP 
Nose r down ( Balanced but negative stiffness \
Figure 2.3 Pitching moment of an airplane about the CG.
Pitch Stiffness
Suppose that the airplane of curve a on Fig. 2.3 is disturbed from its equilibrium attitude, the angle of attack being increased to that at B while its speed remains unal tered. It is now subject to a negative, or nosedown, moment, whose magnitude corre sponds to BC. This moment tends to reduce the angle of attack to its equilibrium value, and hence is a restoring moment. In this case, the airplane has positive pitch stiffness, obviously a desirable characteristic.
On the other hand, if C, were given by the curve b, the moment acting when dis turbed would be positive, or noseup, and would tend to rotate the airplane still far ther from its equilibrium attitude. We see that the pitch stiffness is determined by the sign and magnitude of the slope aC,/aa. I f the pitch stifiess is to be positive at the equilibrium a, C, must be zero, and aCJ& must be negative. It will be appreciated from Fig. 2.3 that an alternative statement is "C,, must be positive, and X,/& neg ative if the airplane is to meet this (limited) condition for stable equilibrium." The various possibilities corresponding to the possible signs of C,, and aC,/aa are shown in Figs. 2.3 and 2.4.
'a
Figure 2.4 Other possibilities.
22 Chapter 2. Static Stability and ControlPart 1
Positive camber Zero camber Neaative camber C,, negative emo ' 0 Cmo positive
Figure 2.5 C,, of airfoil sections.
Possible Configurations
The possible solutions for a suitable configuration are readily discussed in terms of the requirements on C,, and aCJaa. We state here without proof (this is given in Sec. 2.3) that aC,/aa can be made negative for virtually any combination of 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 rather the requirement that the airplane must be simultaneously balanced and have positive pitch stiffness. Since a proper choice of the CG location can ensure a nega tive aCm/aa, then any configuration with a positive C,, can satisfy the (limited) con ditions for balanced and stable flight.
Figure 2.5 shows the C,, of conventional airfoil sections. If an airplane were to consist of a straight wing alone (flying wing), then the wing camber would determine the airplane characteristics as follows:
Negative camberflight possible at a > 0; i.e., C, > 0 (Fig. 2 .3~) .
Zero camberflight possible only at a = 0, or C, = 0. Positive camberflight not possible at any positive a or C,.
For straightwinged tailless airplanes, only the negative camber satisfies the con ditions for stable, balanced flight. Effectively the same result is attained if a flap, de flected upward, is incorporated at the trailing edge of a symmetrical airfoil. A con ventional lowspeed airplane, with essentially straight wings and positive camber, could fly upside down without a tail, provided the CG were far enough forward (ahead of the wing mean aerodynamic center). Flying wing airplanes based on a straight wing with negative camber are not in general use for three main reasons:
+ Cambered wing at CL = 0 Tail with CL negative
Tail with CL positive + Cambered wing at CL = 0
16)
Figure 2.6 Wingtail arrangements with positive C,,,. (a) Conventional arrangement. (b) Tailfirst or canard arrangement.
2.2 Synthesis of Lgt and Pitching Moment 23
+ Lift
v (relative wind)
 Lift
 Lift
Figure 2.7 Sweptback wing with twisted tips.
1. The dynamic characteristics tend to be unsatisfactory.
2. The permissible CG range is too small.
3. The drag and CLmA, characteristics are not good.
The positively cambered straight wing can be used only in conjunction with an auxiliary device that provides the positive C,,,. The solution adopted by experi menters as far back as Samuel Henson (1842) and John Stringfellow (1848) was to add a tail behind the wing. The Wright brothers (1903) used a tail ahead of the wing (canard configuration). Either of these alternatives can supply a positive C,,,, as illus trated in Fig. 2.6. When the wing is at zero lift, the auxiliary surface must provide a noseup moment. The conventional tail must therefore be at a negative angle of at tack, and the canard tail at a positive angle.
An alternative to the wingtail combination is the sweptback wing with twisted tips (Fig. 2.7). When the net lift is zero, the forward part of the wing has positive lift, and the rear part negative. The result is a positive couple, as desired.
A variant of the sweptback wing is the delta wing. The positive C,, can be achieved with such planforms by twisting the tips, by employing negative camber, or by incorporating an upturned tailing edge flap.
2.2 Synthesis of Lift and Pitching Moment
The total lift and pitching moment of an airplane are, in general, functions of angle of attack, controlsurface angle(s), Mach number, Reynolds number, thrust coefficient, and dynamic p re~su re .~ (The lastnamed quantity enters because of aeroelastic ef fects. Changes in the dynamic pressure (ipv2), when all the other parameters are con stant, may induce enough distortion of the structure to alter C, significantly.) An ac curate determination of the lift and pitching moment is one of the major tasks in a static stability analysis. Extensive use is made of windtunnel tests, supplemented by aerodynamic and aeroelastic analyses.
'When partial derivatives are taken in the following equations with respect to one of these variables, for example, aC,,/aa, it is to be understood that all the others are held constant.
24 Chapter 2. Static Stability and ControlPart 1
For purposes of estimation, the total lift and pitching moment may be synthe sized from the contributions of the various parts of the airplane, that is, wing, body, nacelles, propulsive system, and tail, and their mutual interferences. Some data for estimating the various aerodynamic parameters involved are contained in Appendix B, while the general formulation of the equations, in terms of these parameters, fol lows here. In this chapter aeroelastic effects are not included. Hence the analysis ap plies to a rigid airplane.
LIFT AND PITCHING MOMENT OF THE WING
The aerodynamic forces on any lifting surface can be represented as a lift and drag acting at the mean aerodynamic center, together with a pitching couple independent of the angle of attack (Fig. 2.8). The pitching moment of this force system about the CG is given by (Fig. 2.9)3
M, = MaCw + (L, cos a, + D, sin a,)(h  hnw)T
+ (L, sin a,  D, cos a,)z @.%I)
We assume that the angle of attack is sufficiently small to justify the approximations
c o s a w = l , s i n a , = a ,
and the equation is made nondimensional by dividing through by &pV2S?. It then be comes
cmw = cmacw + ( c ~ w + C~waw)(h  hnw) + (CL,~,  C ~ w ) d c (2.22)
Although it may occasionally be necessary to retain all the terms in (2.2,2), experi ence has shown that the last term is frequently negligible, and that CDwaw may be ne glected in comparison with CLw. With these simplifications, we obtain
Mean aerodynamic center
Figure 2.8 Aerodynamic forces on the wing.
3The notation hnw indicates that the mean aerodynamic center of the wing is also the neutral point of the wing. Neutral point is defined in Sec. 2.3.
2.2 Synthesis of Lyt and Pitching Moment 25
Mean Wing zero aerodynamic lift direction
Figure 2.9 Moment about the CG in the plane of symmetry.
where a, = CLcpw is the liftcurveslope of the wing. Equation 2.2,3 will be used to represent the wing pitching moment in the discus
sions that follow.
LIFT AND PITCHING MOMENT OF THE BODY AND NACELLES
The influences of the body and nacelles are complex. A body alone in an airstream is subjected to aerodynamic forces. These, like those on the wing, may be represented over moderate ranges of angle of attack by lift and drag forces at an aerodynamic center, and a pitching couple independent of a. Also as for a wing alone, the lifta re lation is approximately linear. When the wing and body are put together, however, a simple superposition of the aerodynamic forces that act upon them separately does not give a correct result. Strong interference effects are usually present, the flow field of the wing affecting the forces on the body, and vice versa.
These interference flow fields are illustrated for subsonic flow in Fig. 2.10. Part (a) shows the pattern of induced velocity along the body that is caused by the wing vortex system. This induced flow produces a positive moment that increases with wing lift or a. Hence a positive (destabilizing) contribution to Cmm results. Part (b) shows an effect of the body on the wing. When the body axis is at angle a to the stream, there is a crossflow component V sin a . The body distorts this flow locally, leading to crossflow components that can be of order 2V sin a at the wingbody in tersection. There is a resulting change in the wing lift distribution.
The result of adding a body and nacelles to a wing may usually be interpreted as a shift (forward) of the mean aerodynamic center, an increase in the liftcurve slope, and a negative increment in em"<. The equation that corresponds to (2.2,3) for a wing bodynacelle combination is then of the same form as (2.2,3), but with different val ues of the parameters. The subscript wb is used to denote these values.
where a,, is the liftcurveslope of the wingbodynacelle combination.
26 Chapter 2. Static Stability and ControlPart 1
(6) Figure 2.10 Example of mutual interference flow fields of wing and bodysubsonic flow. (a) Qualitative pattern of upwash and downwash induced along the body axis by the wing vorticity. (6) Qualitative pattern of upwash induced along wing by the crossflow past the body.
LIFT AND PITCHING MOMENT OF THE TAIL
The forces on an isolated tail are represented just like those on an isolated wing. When the tail is mounted on an airplane, however, important interferences occur. The most significant of these, and one that is usually predictable by aerodynamic theory, is a downward deflection of the flow at the tail caused by the wing. This is character ized by the mean downwash angle E. Blanking of part of the tail by the body is a sec ond effect, and a reduction of the relative wind when the tail lies in the wing wake is the third.
Figure 2.11 depicts the forces acting on the tail showing the relative wind vector of the airplane. V' is the average or effective relative wind at the tail. The tail lift and drag forces are, respectively, perpendicular and parallel to V'. The reader should note
Figure 2.11 Forces acting on the tail.
p  
2.2 Synthesis of Lift and Pitching Moment 27
the tail angle it, which must be positive as shown for equilibrium. This is sometimes referred to as longitudinal dihedral.
The contribution of the tail to the airplane lift, which by definition is perpendicu lar to V, is
L, cos E  D, sin E
E is always a small angle, and we assume that D,E may be neglected compared with L,. The contribution of the tail to the airplane lift then becomes simply L,. We intro duce the symbol CL, to represent the lift coefficient of the tail, based on the airplane dynamic pressure ipV2 and the tail area S,.
The total lift of the airplane is
L = L,, + L,, 1
or in coefficient form 1 i
The reader should note that the lift coefficient of the tail is often based on the local dynamic pressure at the tail, which differs from ipv2 when the tail lies in the wing wake. This practice entails carrying the ratio Vr/V in many subsequent equations. The definition employed here amounts to incorporating V'IV into the tail liftcurve slope a,. This quantity is in any event different from that for the isolated tail, owing to the interference effects previously noted. This circumstance is handled in various ways in the literature. Sometimes a tail efficiency factor 7, is introduced, the isolated tail lift slope being multiplied by 7,. In other treatments, 7, is used to represent ( v ' / v ) ~ . In the convention adopted here, a, is the liftcurve slope of the tail, as measured in situ on the airplane, and based on the dynamic pressure hpV2. This is the quantity that is directly obtained in a windtunnel test.
From Fig. 2.1 1 we find the pitching moment of the tail about the CG to be
M, = l,[L, cos (a,,  E) + D, sin (a,,  E ) ]
 zt[Dr cos (a,,  E)  L, sin (a,,  E ) ] + Mu,., (2.2,7)
Experience has shown that in the majority of instances the dominant term in this equation is the first one, and that all others are negligible by comparison. Only this case will be dealt with here. The reader is left to extend the analysis to cases in which this approximation is not valid. With the above approximation, and that of small an gles,
M , = 1,L, =  ~ , C , ~ ~ V ~ S ,
Upon conversion to coefficient form, we obtain
The combination l tS,IS~ is the ratio of two volumes characteristic of the airplane's
28 Chapter 2. Static Stability and ControlPart I
mean aerodynamic
Tail mean 7' aerodynamic
P h n c u b F l I I center
Figure 2.12 Wingbody and tail mean aerodynamic centers.
geometry. It is commonly called the "horizontaltail volume ratio," or more simply, the "tail volume." It is denoted here by V,. Thus
Cm, = VHCL, ( 2 . 2 3
Since the center of gravity is not a fixed point, but varies with the loading condi tion and fuel consumption of the vehicle, VH in (2.2,9) is not a constant (although it does not vary much). It is a little more convenient to calculate the moment of the tail about a fixed point, the mean aerodynamic center of the wingbody combination, and to use this moment in the subsequent algebraic manipulations. Figure 2.12 shows the relevant relationships, and we define
which leads to
The moment of the tail about the wingbody mean aerodynamic center is then [cf. (2.2,9)1
and its moment about the CG is, from substitution of (2.2,11) into (2.2,9)
PITCHING MOMENT OF A PROPULSIVE SYSTEM
The moment provided by a propulsive system is in two parts: (1) that coming from the forces acting on the unit itself, for example, the thrust and inplane force acting on a propeller, and (2) that coming from the interaction of the propulsive slipstream with the other parts of the airplane. These are discussed in more detail in Sec. 3.4. We assume that the interference part is included in the moments already given for the wing, body, and tail, and denote by Cmp the remaining moment from the propulsion units.
2.3 Total Pitching Moment and Neutral Point 29
2.3 Total Pitching Moment and Neutral Point
On summing the first of (2.2,4) and (2.2,13) making use of (2.2,6) and adding the contribution Cmp for the propulsive system, we obtain the total pitching moment about the CG
It is worthwhile repeating that no assumptions about thrust, compressibility, or aero elastic effects have been made in respect of (2.3,1). The pitch stiffness ( CmU) is now obtained from (2.3,l). Recall that the mean aerodynamic centers of the wingbody combination and of the tail are fixed points, so that
If a true mean aerodynamic center in the classical sense exists, then aCmy~whlaa is zero and
 acLt ac, Cmn = CLU(h  h,J  VH  + a& a a
CmU as given by (2.3,2) or (2.3,3) depends linearly on the CG position, h. Since CLU is usually large, the magnitude and sign of Cmo depend strongly on h. This is the basis of the statement in Sec. 2.2 that CmU can always be made negative by a suitable choice of h. The CG position h, for which CmU is zero is of particular significance, since this represents a boundary between positive and negative pitch stiffness. In this book we define h, as the neutral point, NP. It has the same significance for the vehi cle as a whole as does the mean aerodynamic center for a wing alone, and indeed the term vehicle aerodynamic center is an acceptable alternative to "neutral point."
The location of the NP is readily calculated from (2.3,2) by setting the lefthand side to zero leading to
Substitution of (2.3,4) back into (2.3,2) simplifies the latter to
which is valid whether Cmll(,",, and C,,, vary with a or not. Equation (2.3,5) clearly provides an excellent way of finding h, from test results, that is from measurements of Cmcr and CLU. The difference between the CG position and the NP is sometimes called the static margin,
K, = (h,  h) (2.36)
Since the criterion to be satisfied is C,,,= < 0, that is, positive pitch stiffness, then we see that we must have h < h,, or K, > 0. In other words the CG must be forward of the NP. The farther forward the CG the greater is K,, and in the sense of "static stability" the more stable the vehicle.
The neutral point has sometimes been defined as the CG location at which the derivative dCmldCL = 0. When this definition is applied to the gliding flight of a rigid
30 Chapter 2. Static Stability and ControlPart I
airplane at low Mach number, the neutral point obtained is identical with that defined in this book. This is so because under these restricted conditions CL is a unique func tion of a, and dCmldCL = (aC,laa)l(aC,laa). Then dCmldCL and aCm/aa are simul taneously zero. In general, however, C, and CL are both functions of several vari ables, as pointed out at the beginning of Sec. 2.2. For fixed values of 6, and h, and neglecting Reynolds number effects (these are usually very small), we may write
where C, is the thrust coefficient, defined in Sec. 3.15. Mathematically speaking, the derivative dC,ldC, does not exist unless M, C ,
and ipV2 are functions of CL. When that is the case, then
dc, ac,,, aa ac, aM ac, ac, ac, a ( $ p ~ 2 )  + ++ dCL aa acL aM ac, ac, acL a(apv2) acL (2.3,8)
Equation 2.3,8 has meaning only when a specific kind of flight is prescribed: e.g., horizontal unaccelerated flight, or rectilinear climbing flight at full throttle. When a condition of this kind is imposed, then M, C , and the dynamic pressure are definite functions of CL, dC,JdCL exists, and a neutral point may be calculated. The neutral point so found is not an index of stability with respect to angle of attack disturbances, and the question arises as to what it does relate to. It can be shown that it relates to the trim curves of the airplane. A plot of the elevator angle to trim versus speed will have a zero slope when dCmldCL is zero, and a negative slope when the CG lies aft of the neutral point so defined. As shown in Sec. 2.4, this reversal of slope indicates a tendency toward instability with respect to speed, but only a dynamic analysis can show whether or not the airplane is stable in this condition. There are cases when the application of the "trimslope" criterion can be definitely misleading as to stability. One such is level unaccelerated flight, during which the throttle must be adjusted every time the flight speed or C, is altered.
It can be seen from the foregoing remarks that the "trimslope" criterion for the neutral point does not lead to any definite and clearcut conclusions, either about the stability with respect to angle of attack disturbances, or about the general static sta bility involving both speed and angle of attack disturbances. It is mainly for this rea son that the neutral point has been defined herein on the basis of aCJaa.
EFFECT OF LINEAR LIFT AND MOMENT ON NEUTRAL POINT
When the forces and moments on the wing, body, tail, and propulsive system are lin ear in a, as may be near enough the case in reality, some additional useful relations can be obtained. We then have
and
If Cmwb is linear in CLw,, it can be shown (see exercise 2.3) that Cmacwb does not vary with C,,, i.e. that a true mean aerodynamic center exists. Figure 2.1 1 shows that the tail angle of attack is
2.3 Total Pitching Moment and Neutral Point 31
at = a,,  it  E (2.3,12)
and hence CL, = a,(awb  it  E ) (2.3,13)
The downwash E can usually be adequately approximated by
The downwash E, at a,, = 0 results from the induced velocity field of the body and from wing twist; the latter produces a vortex wake and downwash field even at zero total lift. The constant derivative adaa occurs because the main contribution to the downwash at the tail comes from the wing trailing vortex wake, the strength of which is, in the linear case, proportional to CL.
The tail lift coefficient then is
CL, = a, a,, 1    i  E [ ( ) "1 and the total lift, from (2.2,6) and (2.3,9) is
or since a,, and a differ by a constant
where
is the coefficient of the lift on the tail when a,, = 0;
is the liftcurve slope of the whole configuration; and a is the angle of attack of the zerolift line of the whole configuration (see Fig. 2.13). Note that, since it is positive,
Figure 2.13 Graph of total lift.
32 Chapter 2. Static Stability and ControlPart I
then (C,), is negative. The difference between a and a,, is found by equating (2.3,16b and c) to be
When the linear relations for C,, C,, and Cmp are substituted into (2.3,l) the fol lowing results can be obtained after some algebraic reduction:
where Cm, = a(h  h,,)  a (a)
(2.3,21)
or Cm, = a,,(h  h,3  (b)
and em, = CmaC, + Cmop + a,%(€o + it)
 acmp where Cmop = CmOp + ( a  a,,)  aa
Note that since cm, is the pitching moment at zero a,,, not at zero total lift, its value depends on h (via VH), whereas C,,, being the moment at zero total lift, represents a couple and is hence independent of CG position. All the above relations apply to tail less aircraft by putting pH = 0. Another useful relation comes from integrating (2.3,5), i.e.
Cm = Cm, + C,(h  hn) (a) Cm = Cm, + aa(h  h,) (b) (2.3,25)
Cma = a(h  h,) (c)
Figure 2.14 shows the linear Cm vs. a relation, and Fig. 2.15 shows the resultant sys tem of lift and moment that corresponds to (2.3,25), that is a force C, and a couple
Figure 2.14 Effect of CG location on C,,, curve.
2.4 Longitudinal Control 33
Figure 2.25 Total lift and moment acting on vehicle.
C,,, at the NP. Figure 2.15 is a very important result that the student should fix in his mind.
2.4 Longitudinal Control
In this section we discuss the longitudinal control of the vehicle from a static point of view. That is, we concern ourselves with how the equilibrium state of steady rectilin ear flight is governed by the available controls. Basically there are two kinds of changes that can be made by the pilot or automatic control systema change of propulsive thrust, or a change of configuration. Included in the latter are the opera tion of aerodynamic controlselevators, wing flaps, spoilers, and horizontal tail ro tation. Since the equilibrium state is dominated by the requirement Cm = 0, the most powerful controls are those that have the greatest effect on C,.
Figure 2.14 shows that another theoretically possible way of changing the trim condition is to move the CG, which changes the value of a at which C, = 0. Moving it forward reduces the trim a or C,, and hence produces an increase in the trim speed. This method was actually used by Lilienthal, a pioneer of aviation, in gliding flights during 189111896, in which he shifted his body to move the CG. It has the inherent disadvantage, apart from practical difficulties, of changing Cmcx at the same time, re ducing the pitch stiffness and hence stability, when the trim speed is reduced.
The longitudinal control now generally used is aerodynamic. A variable pitching moment is provided by moving the elevator, which may be all or part of the tail, or a trailingedge flap in a tailless design. Deflection of the elevator through an angle 6, produces increments in both the Cm and CL of the airplane. The AC, caused by the el evator of aircraft with tails is small enough to be neglected for many purposes. This is not so for tailless aircraft, where the AC, due to elevators is usually significant. We shall assume that the lift and moment increments for both kinds of airplane are linear in a,, which is a fair representation of the characteristics of typical controls at high Reynolds number. Therefore,
and cm = Cm(ff> + Cm,6, (4
where C,se = aCLla6,, CmSe = aC,,,la8,, and CL(a), Cm(a) are the "basic" lift and mo ment when 6, = 0. The usual convention is to take down elevator as positive (Fig. 2.16a). This leads to positive C L , and negative C,,,,. The deflection of the elevator
34 Chapter 2. Static Stability and ControlPart 1
,Horizontal tail
Original trim a
0 (4
Figure 2.16 Effect of elevator angle on C, curve. (a) Elevator angle. (b) C,,,  a curve. (c ) C,  a curve.
through a constant positive angle then shifts the Cma curve downward, without change of slope (Fig. 2.16b). At the same time the zerolift angle of the airplane is slightly changed (Fig. 2.16~).
In the case of linear lift and moment, we have
THE DERIVATIVES C,, AND Cm6
Equation (2.2,6) gives the vehicle lift. Hence
acL  JCL,", st acL, CLg, =   +   as, as, ~ a s ,
2.4 Longitudinal Control 35
in which only the first term applies for tailless aircraft and the second for conven tional tail elevators or all moving tails (when i, is used instead of 6,). We define the elevator lift effectiveness as
so that (2.4,3) becomes
and the lift coefficient of the tail is
CL, = atat + a,&. (b)
The total vehicle Cm is given for both tailed and tailless types by (2.3,l). For the latter, of course, VH = 0. Taking the derivative with respect to 8, gives
We may usually neglect the last term, since there is unlikely to be any propulsiveele vator interaction that cannot be included in a,. Then (2.4,6) becomes
  acme<.,,, C m ~ as, + CL,<,(h 
Summarizing for both types of vehicle, we have (retaining only the dominant terms) Tailed aircraft:
Cm.se = aeVH + CL8,,(h  hnw,)
Tailless aircraft:
In the last case, the subscript wb is, of course, redundant and has been dropped. The primary parameters to be predicted or measured are a, for tailed aircraft, and dCL/aS,, aCmz,,la6, for tailless.
ELEVATOR ANGLE TO TRIM
The trim condition is C, = 0, whence from (2.4,ld)
36 Chapter 2. Static Stability and ControlPart 1
and the corresponding lift coefficient is
When the linear lift and moment relations (2.4,2) apply the equations for trim are
These equations are solved for a and 6, to give
CrnoC~g~ + Crng,C~unm atrim =
det
  CrnOCL, + Crn,CL,",,, set,,  det
d'etnm   Cma CL,     det det (h  h,) (c)
dc~tnm where det = C L ~ C ~ ,  C,,Crn, (d )
is the determinant of the square matrix in (2.4,12) and is normally negative. The val ues of det for the two types of airplane are readily calculated from (2.4,8 and 9) to gether with (2.3,5) to give
Tailed aircraft:
det = CL,[CLg,(hn  hnwb)  aevHl (a>
Tailless aircraft:
acmac det = C,,  a
and both are independent of h. From (2.4,13a) we get the trimmed lift curve:
and the slope is given by
The trimmed liftcurve slope is seen to be less than CLa by an amount that depends on C,,, i.e., on the static margin, and that vanishes when h = h,. The difference is only a few percent for tailed airplanes at normal CG position, but may be appreciable for tailless vehicles because of their larger C,,. The relation between the basic and trimmed lift curves is shown in Fig. 2.17.
Equation (2.4,13b) is plotted on Fig. 2.18, showing how 6e,nm varies with Chnm and CG position when the aerodynamic coefficients are constant.
2.4 Longitudinal Control 37
Figure 2.17 Trimmed lift curve.
VARIATION OF Setrim WITH SPEED
When, in the absence of compressibility, aeroelastic effects, and propulsive system effects, the aerodynamic coefficients of (2.4,13) are constant, the variation of with speed is simple. Then 6,tr,,n is a unique function of CLInm for each CG position. Since CLtrnn, is in turn fixed by the equivalent airspeed: for horizontal flight
then 6,,,_ becomes a unique function of V,. The form of the curves is shown in Fig. 2.19 for representative values of the coefficients.
The variation of 6etr,m with CLmm or speed shown on Figs. 2.18 and 2.19 is the normal and desirable one. For any CG position, an increase in trim speed from any initial value to a larger one requires a downward deflection of the elevator (a forward
Figure 2.18 Elevator angle to trim at various CG positions.
4Equivalent airspeed (EAS) is V, = V where p, is standard sealevel density.
38 Chapter 2. Static Stability and ControlPart 1
4
L
Figure 2.19 Example of variation of elevator angle to trim with speed and CG position.
movement of the pilot's control). The "gradient" of the movement a8etnm/aVE is seen to decrease with rearward movement of the CG until it vanishes altogether at the NP. In this condition the pilot in effect has no control over trim speed, and control of the vehicle becomes very difficult. For even more rearward positions of the CG the gra dient reverses, and the controllability deteriorates still further.
When the aerodynamic coefficients vary with speed, the above simple analysis must be extended. In order to be still more general, we shall in the following explic itly include propulsive effects as well, by means of the parameter a,, which stands for the state of the pilot's propulsion control (e.g., throttle position). 6, = constant there fore denotes fixedthrottle and, of course, for horizontal flight at varying speed, 6, must be a function of V that is compatible with T = D. For angles of climb or descent in the normal range of conventional airplanes L = W is a reasonable approximation, and we adopt it in the following. When nonhorizontal flight is thus included, 6, be comes an independent variable, with the angle of climb y then becoming a function of a,, V, and altitude.
The two basic conditions then, for trimmed steady flight on a straight line are
and in accordance with the postulates made above, we write
Now let ( ), denote one state that satisfies (2,4,18) and consider a small change from it, denoted by differentials, to another such state. From (2.4,18) we get, for p = const,
dCm = 0 (2.4,20)
and CLV2 = const
or 2VeCL$V + VidCL = 0
dV so that dCL = 2CLe  = 2CLedV
ve
2.4 Longitudinal Control 39
where 9 is VN, . Taking the differentials of (2.4,19) and equating to (2.4,20 and 21) we get
CLmda + CLaed6, = CL,d6,  (CLv + 2cLe)dP (2.4,22)
Cmmda + C,,d& = CmaPd$  C,,di/
where C,, = acLla9 and C,,,, = aC,lao. From (2.4,22) we get the solution for d6, as
[(C,, + 2CL,)C,,  ~ L ~ c , , l d o + (CL,C,,,,,  CL,C,,)dS, det
There are two possibilities, a,, constant and 6, variable. In the first case (fixed throt tle), dS, = 0 and
(C,, + 2CLe)Cmrn  CL,Cmv det
It will be shown in Chap. 6 that the vanishing of this quantity is a true criterion of stability, that is it must be >O for a stable airplane. In the second case, for example exactly horizontal flight, 6, = 6,(V) and the $term on the righthand side of (2.4,23) remains. For such cases the gradient (d6,,r,mldV) is not necessarily related to stability. For purposes of calculating the propulsion contributions, the terms CL,p d6, and C,,,, d6, in (2.4,23) would be evaluated as dCLp and dCmP [see the notation of (2.3,1)]. These contributions to the lift and moment are discussed in Sec. 3.4.
The derivatives CLv and C,, may be quite large owing to slipstream effects on STOL airplanes, aeroelastic effects, or Mach number effects near transonic speeds. These variations with M can result in reversal of the slope of as illustrated on Fig. 2.20. The negative slope at A, according to the stability criterion referred to above, indicates that the airplane is unstable at A. This can be seen as follows. Let the airplane be in equilibrium flight at the point A, and be subsequently perturbed so that its speed increases to that of B with no change in a or 8,. Now at B the elevator angle is too positive for trim: that is there is an unbalanced nosedown moment on the air plane. This puts the airplane into a dive and increases its speed still further. The speed will continue to increase until point C is reached, when the 6, is again the correct value for trim, but here the slope is positive and there is no tendency for the speed to change any further.
Figure 2.20 Reversal of slope at transonic speeds, 6, = const.
40 Chapter 2. Static Stability and ControlPart I
STATIC STABILITY LIMIT, h,
The critical CG position for zero elevator trim slope (i.e. for stability) can be found by setting (2.4,24) equal to zero. Recalling that Cma = CL,(h  h,), this yields
where
Depending on the sign of Cmv, h, may be greater or less than h,. In terms of h,, (2.4,24) can be rewritten as
(h  h,) is the "stability margin," which may be greater or less than the static margin.
FLIGHT DETERMINATION OF h, AND h,
For the general case, (2.3,5) suggests that the measurement of h, requires the mea surement of Cma and CLa. Flight measurements of aerodynamic derivatives such as these can be made by dynamic techniques. However, in the simpler case when the complications presented by propulsive, compressibility, or aeroelastic effects are ab sent, then the relations implicit in Figs. 2.18 and 2.19 lead to a means of finding h, from the elevator trim curves. In that case all the coefficients of (2.4,13) are con stants, and
dSettim  'ma    d C ~ t i , det
d'emim  C L ~    det (h  h,)
dc~rxm
Thus measurements of the slope of 6e,r,m VS. Ck, at various CG positions produce a 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 is clear from (2.4,27) that a plot of (d6eltimld~sp against h will determine h, as the point where the curve crosses the h axis.
M fixed
Figure 2.21 Determination of stickfixed neutral point from flight test.
2.5 The Control Hinge Moment 41
2.5 The Control Hinge Moment
To rotate any of the aerodynamic control surfaces, elevator, aileron, or rudder, about its hinge, it is necessary to apply a force to it to overcome the aerodynamic pressures that resist the motion. This force may 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 from the control surface ("flybywire" or "flybylight"). In any case, the force that has to be applied to the control surface must be known with precision if the control system that connects the primary controls in the cockpit to the aerodynamic surface is to be de signed correctly. The range of control system options is so great that it is not feasible in this text to present a comprehensive coverage of them. We have therefore limited ourselves in this and the following chapter to some material related to elevator con trol forces when the human pilot supplies all of the actuation, or when a power assist relieves the pilot of a fixed fraction of the force required. This treatment necessarily begins with a discussion of the aerodynamics; that is, of the aerodynamic hinge mo ment.
The aerodynamic forces on any control surface produce a moment about the hinge. Figure 2.22 shows a typical tail surface incorporating an elevator with a tab. The tab usually exerts a negligible effect on the lift of the aerodynamic surface to which it is attached, although its influence on the hinge moment is large.
\  Tailplane
Elevator
I I  I . ,  1
Tab hinge
Elevator hinge
Tab hinge
Figure 2.22 Elevator and tab geometry. (a ) Plan view. (b) Section AA.
42 Chapter 2. Static Stability and ControlPart 1
The coefficient of elevator hinge moment is defined by
Here H, is the moment, about the elevator hinge line, of the aerodynamic forces on the elevator and tab, S, is the area of that portion of the elevator and tab that lies aft of the elevator hinge line, and Z, is a mean chord of the same portion of the elevator and tab. Sometimes ?, is taken to be the geometric mean value, that is, ?, = SJ2se, and other times it is the rootmean square of c,. The taper of elevators is usually slight, and the difference between the two values is generally small. The reader is cautioned to note which definition is employed when using reports on experimental measure ments of Che.
Of all the aerodynamic parameters required in stability and control analysis, the hingemoment coefficients are most difficult to determine with precision. A large number of geometrical parameters influence these coefficients, and the range of de sign configurations is wide. Scale effects tend to be larger than for many other pa rameters, owing to the sensitivity of the hinge moment to the state of the boundary layer at the trailing edge. Twodimensional airfoil theory shows that the hinge mo ment of simple flap controls is linear with angle of attack and control angle in both subsonic and supersonic flow.
The normalforce distributions typical of subsonic flow associated with changes in a and 6, are shown qualitatively in Fig. 2.23. The force acting on the movable flap has a moment about the hinge that is quite sensitive to its location. Ordinarily the hinge moments in both cases (a) and (b) shown are negative.
In many practical cases it is a satisfactory engineering approximation to assume that for finite surfaces Che is a linear function of as, 6,, and 6,. The reader should note however that there are important exceptions in which strong nonlinearities are present.
We assume therefore that Che is linear, as follows,
where
a, is the angle of attack of the surface to which the control is attached (wing or tail), and 6, is the angle of deflection of the tab (positive down). The determination of the hinge moment then resolves itself into the determination of b,, b,, b,, and b,. The geometrical variables that enter are elevator chord ratio cJc,, balance ratio cdc,, nose shape, hinge location, gap, trailingedge angle, and planform. When a setback hinge is used, some of the pressure acts ahead of the hinge, and the hinge moment is less than that of a simple flap with a hinge at its leading edge. The force that the control system must exert to hold the elevator at the desired angle is in direct proportion to the hinge moment.
2.5 The Control Hinge Moment 43
Figure 2.23 Normalforce distribution over control surface at subsonic speed. (a ) Force distribution over control associated with a, at 8, = 0. (b) Force distribution over control associated with 6, at zero a,.
We shall find it convenient subsequently to have an equation like (2.5,l) with a instead of a,7. For tailless aircraft, a, is equal to a, but for aircraft with tails, a, = a,. Let us write for both types
where for tailless aircraft C,,,, = b,, Chea = b, . For aircraft with tails, the relation be tween a and at is derived from (2.3,12) and (2.3,19), that is,
whence it follows that for tailed aircraft, with symmetrical airfoil sections in the tail, for which b, = 0,
44 Chapter 2. Static StabiZiQ and [emailprotected] 1
2.6 Influence of a Free Elevator on Lift and Moment In Sec. 2.3 we have dealt with the pitch stiffness of an airplane the controls of which are fixed in position. Even with a completely rigid structure, which never exists, a manually operated control cannot be regarded as fixed. A human pilot is incapable of supplying an ideal rigid constraint. When irreversible power controls are fitted, how ever, the stickfixed condition is closely approximated. A characteristic of interest from the point of view of handling qualities is the stability of the airplane when the elevator is completely free to rotate about its hinge under the influence of the aerody namic pressures that act upon it. Normally, the stability in the controlfree condition is less than with fixed controls. It is desirable that this difference should be small. Since friction is always present in the control system, the free control is never real ized in practice either. However, the two ideal conditions, free control and fixed con trol, represent the possible extremes.
When the control is free, then Ch, = 0, so that from (2.5,2)
The typical upward deflection of a freeelevator on a tail is shown in Fig. 2.24. The corresponding lift and moment are
After substituting (2.6,l) into (2.6,2), we get
where
When due consideration is given to the usual signs of the coefficients in these equa tions, we see that the two important gradients CLa and Cma are reduced in absolute
Figure 2.24 Elevator floating angle.
2.6 Influence of a Free Elevator on Lift and Moment 45
magnitude when the control is released. This leads, broadly speaking, to a reduction of stability.
FREEELEVATOR FACTOR
When the elevator is free, the liftcurve slope is given by (2.6,4b), that is,
The factor in parentheses is the free elevator factol; and normally has a value less than unity. When the elevator is part of the tail, the floating angle can be related to a,, viz for b, = 0
and the tail lift coefficient is
"L, = arat + aeaefmr
The effective liftcurve slope is
acL,   Fa, 3%
where F = 1    is the free elevator factor for a tail. If Fa, is used in place ( of a, and a' in place of a, then all the equations given in Sec. 2.3 hold for aircraft with a free elevator.
ELEVATORFREE NEUTRAL POINT
It is evident from the preceding comment that the NP of a tailed aircraft when the el evator is free is given by (2.3,23) as
Fa,  h:, = hnw,, + , V,
a
Alternatively, we can derive the NP location from (2.6,5b), for we know from (2.3~5) that
46 Chapter 2. Static Stability and ControlPart 1
Since C,, is of different form for the two main types of aircraft, we proceed sepa rately below.
TAILED AIRCRAFT
C,, is given by (2.4,8), so (2.6,11) becomes for this case
Using (2.6,4b) this becomes
We replace hnw, by (hnW,  h,) + h, to get
cheacLs, aeChea  h:, = h, +
b2ar ( h n  hnwJ   arb2 VH
Finally, using (2.4,8) for CL,, and (2.5,4) for Chea, we get
TAILLESS AIRCRAFT
C,, is given by (2.4,9) and Chea = b,. When these are substituted into (2.6,11) the re sult is
By virtue of (2.6,6) this becomes
The difference (h:  h) is called the controlfree static margin, KL. When representa tive numerical values are used in (2.6,13) one finds that h,  hl, may be typically about 0.08. This represents a substantial forward movement of the NP, with conse quent reduction of static margin, pitch stiffness, and stability.
2.7 The Use of Tabs 47
2.7 The Use of Tabs
TRIM TABS
In order to fly at a given speed, or CL, it has been shown in Sec. 2.4 that a certain ele vator angle is required. When this differs from the freefloating angle a force is required to hold the elevator. When flying for long periods at a constant speed, it is very fatiguing for the pilot to maintain such a force. The trim tabs are used to relieve the pilot of this load by causing a,,,, and to coincide. The trim tab angle required is calculated below.
When C,, and Cm are both zero, the tab angle is obtained from (2.5,2) as
1 aftr,m =   (Cheo + Che,atr~rn + b2Set,,m) (2.7,1)
b3
On substituting from (2.4,13) (which implies neglecting aCm/a8,), we get
which is linear in Ckr,_ for constant h, as shown in Fig. 2.25. The dependence on h is simple, since from (2.6,11) we find that
and hence
1 cm(, a'b, at,r,,,, =   + (Che,cL&e  b2CLa)   det
(h  h:)C,r,m (2.792) 63 I
This result applies to both tailed and tailless aircraft, provided only that the appropri ate values of the coefficients are used. It should be realized, of course, in reference to Fig. 2.25, that each different CLtrim in a real flight situation corresponds to a different set of values of M, ipV2, and C , so that in general the coefficients of (2.7,2) vary with CL, and the graphs will depart from straight lines.
Figure 2.25 Tab angle to trim.
48 Chapter 2. Static Stability and ControlPart 1
Equation (2.7,2) shows that the slope of the vs Ck", curve is proportional to the controlfree static margin. When the coefficients are constants, we have
d6rmm  b2 a'    dC,,, bs det (h  h;)
The similarity between (2.7,3) and (2.4,13c) is noteworthy, that is the trimtab slope bears the same relation to the controlfree NP as the elevator angle slope does to the controlfixed NP. It follows that flight determination of h: from measurements of dG,,,,ldCktim is possible subject to the same restrictions as discussed in relation to the measurement of h, in Sec. 2.4.
OTHER USES
Tabs are used for purposes other than trimming, especially for manually actuated controls. Three of the main types are as follows:
Geared Tabs. Tabs connected to the main surface by a mechanical linkage that causes the tab to deflect automatically when the main surface is deflected, but in the opposite direction. The hinge moment produced by the tab then assists the rotation of the main surface. These have the effect of reducing the b, of the surface.
Spring Tabs. Tabs connected to the main surface by an elastic element. The design is such that the deflection of the tab depends on the dynamic pressure in a way that mitigates the effect of the speedsquared law on the control force.
Servo Tabs. An arrangement in which the pilot controls the tab directly through a mechanical linkage. It is then the tab, not the pilot, that provides the hinge moment needed to rotate the main surface.
Both spring tabs and servo tabs are effective devices for reducing control forces on large highspeed airplanes. However, both add an additional degree of freedom to the control system dynamics, and this is a potential source of trouble due to vibration or flutter.
For further details of how these tabs function, see Etkin (1972).
2.8 Control Force to Trim
The importance of control forces in relation to handling qualities has already been emphasized in previous sections, and the many options available to designers of pow ered control systems has been noted. Cockpit devices can of course be designed to produce more or less any desired synthetic feel on the primary flight controls. It is nevertheless both instructive and necessary to be able to calculate the control forces that will be present in the case of natural feel, or when a simple power assist is pres ent. A case in point is the elevator force required to trim the airplane, and how it varies with flight speed.
Figure 2.26 is a schematic representation of a reversible control system. The box denoted "control system linkage" represents any assemblage of levers, rods, pulleys,
2.8 Control Force to Trim 49
P, 8
r r I Controlsystem linkage L  J
Figure 2.26 Schematic diagram of an elevator control system.
cables, and powerboost elements that comprise a general control system. We assume that the elements of the linkage and the structure to which it is attached are ideally rigid, so that no strain energy is stored in them, and we neglect friction. The system then has one degree of freedom. P is the force applied by the pilot, (positive to the rear) s is the displacement of the hand grip, and the work done by the power boost system is W,. Considering a small quasistatic displacement from equilibrium (i.e., no kinetic energy appears in the control system), conservation of energy gives
Pds + dW, + H,d6, = 0 (2.8,1)
Now the nature of ratio or power boost controls is such that dWJds is proportional to P or He. Hence we can write
P = (G ,  G,)H, (2.8,2)
where d6,
G I =   > 0, the elevator gearing (radft or radm) ds
and dW&s
G2 =  , the boost gearing (ftpl or m') He
(2.8,2) is now rewritten as
P = GH, (2.8,3)
where G = GI  G,. For fixed G I , i.e., for a given movement of the control su$ace to result from a given displacement of the pilot's control, then the introduction of power boost is seen to reduce G and hence P. G may be designed to be constant over the whole range of 6,, or it may, by the use of special linkages and power systems, be made variable in almost any desired manner.
Introduction of the hingemoment coefficient gives the expression for P as
P = GC,,S,~,&~V~ (2.8,4)
and the variation of P with flight speed depends on both V 2 and on how C,, varies with speed.
50 Chapter 2. Static Stability and ControlPart I
The value of Ch, at trim for arbitrary tab angle is given by
Che = Cheo + Che,atrim + b28etri, + b38t (2.8,5)
(2.8,5) in combination with (2.7,l) yields
the = b3(4  6tm,,> (2.85)
i.e., the hinge moment is zero when 6, = 8,,,, as expected, and linearly proportional to the difference. From (2.7,2) then the hinge moment is
Lift equals the weight in horizontal flight, so that
where w = WIS is the wing loading. When (2.8,7 and 8) are substituted into (2.8,4) the result obtained is
P = A + B $ ~ v ~ (2.89)
where
arb2 A = GSec,w  (h  h:)
det
Cmo b38t + che0 +  det (che,cb  ~ ~ C L J ]
The typical parabolic variation of P with V when the aerodynamic coefficients are all constant, is shown in Fig. 2.27. The following conclusions may be drawn.
Figure 2.27 Example of lowspeed control force.
2.9 Control Force Gradient 51
1. Other things remaining equal, P = S$,, i.e., to the cube of the airplane size. This indicates a very rapid increase in control force with size.
2. P is directly proportional to the gearing G.
3. The CG position only affects the constant term (apart from a secondorder in fluence on C,,). A forward movement of the CG produces an upward transla tion of the curve.
4. The weight of the airplane enters only through the wing loading, a quantity that tends to be constant for airplanes serving a given function, regardless of weight. An increase in wing loading has the same effect as a forward shift of the CG.
5. The part of P that varies with 4pv2 decreases with height, and increases as the speed squared.
6. Of the terms contained in B, none can be said in general to be negligible. All of them are "builtin" constants except for 8,.
7. The effect of the trim tab is to change the coefficient of ipV2, and hence the curvature of the parabola in Fig. 2.27. Thus it controls the intercept of the curve with the V axis. This intercept is denoted V,,,; it is the speed for zero control force.
2.9 Control Force Gradient
It was pointed out in Sec. 2.7 how the trim tabs can be used to reduce the control force to zero. A significant handling characteristic is the gradient of P with Vat P = 0. The manner in which this changes as the CG is moved aft is illustrated in Fig. 2.28. The trim tab is assumed to be set so as to keep V,,, the same. The gradient dP/dV is seen to decrease in magnitude as the CG moves backward. When it is at the controlfree neutral point, A = 0 for aircraft with or without tails, and, under the stated conditions, the PIV graph becomes a straight line lying on the V axis. This is an important characteristic of the controlfree NP; that is, when the CG is at that point, no force is required to change the trim speed.
A quantitative analysis of the controlforce gradient follows.
Figure 2.28 Effect of CG location on controlforce gradient at fixed trim speed.
52 Chapter 2. Static Stability and ControlPart I
The force is given by (2.8,9). From it we obtain the derivative
ap
av  = BpV
At the speed V,,,, P = 0, and B = AI~pV2t,irn, whence
a p 2A
A is given following (2.8,9). Substituting the value into (2.9,l) we get
a p a'b, w
a v  = 2GSece   det V,,, (h 
From (2.9,2) we deduce the following:
1. The controlforce gradient is proportional to Sei',; that is, to the cube of air plane size.
2. It is inversely proportional to the trim speed; i.e. it increases with decreasing speed. This effect is also evident in Fig. 2.27.
3. It is directly proportional to wing loading. 4. It is independent of height for a given true airspeed, but decreases with height
for a fixed V,. 5. It is directly proportional to the controlfree static margin.
Thus, in the absence of Mach number effects, the elevator control will be "heaviest" at sealevel, lowspeed, forward CG, and maximum weight.
2.10 Exercises
2.1 A subsonic transport aircraft has a tapered, untwisted sweptback wing with straight leading and trailing edges. The wing tips are straight and parallel to the root chord. In the following, use the data of Appendix C and assume that the airfoil section local aerodynamic center is at the &chord point.
(a) Make an accurate threeview drawing of the wing chord plane.
(b) Calculate wing area S, aspect ratio A, taper ratio A = c,lc, and the mean aerody namic chord c.
(c) Calculate the location of the wing's mean aerodynamic center, and locate it and c on the side view of the wing (with dimensions). (Assume a uniform additional lift coefficient C," = C,.)
(d) The aircraft is to be operated with its most rearward CG position limited to 25 ft (7.62 m) aft of the apex of the wing. The distance between the wing and tail mean aerodynamic centers is it = 55 ft (16.76 m). Estimate the tail area required to provide a controlfixed static margin of at least 0.05 at all times. Assume that a, = a,, and hnw = h ,w,. Ignore power plant effects and use adaa = 0.25.
Geometric Data
Wing Span, b
Root Chord, c,
Tip Chord, c,
2.10 Exercises 53
Leading edge sweep, A, 26"
Dihedral angle, y 4"
2.2 Evaluate the validity of the approximation made in going from (2.2,2) to (2.2,3) by using the data for the airplane of Exercise (2.1) and calculating Cms from both equa tions. Assume CDw = CD,n,nw + KC;_ and L = L,. The following additional data are provided.
Geometric Data
Weight, W 207,750 lb (924,488 N)
dZ 0.15
Aerodynamic Data
a, 0.080ldeg
cmoc.  0.05
C~mbn ,v 0.013 K 0.054
V 350 kts (1 80 m/s)
P 2.377 X lo' slugs/ft3 (1.225 kg/m3)
2.3 Show that if em_, is a linear function of C,_, then Cm,,,, is a constant.
2.4 Beginning with (2.3,1) perform the reductions to derive (2.3,20) to (2.3,23).
2.5 The following data apply to a scale wind tunnel model of a transport airplane. The fullscale mass of the aircraft is 1,552.80 slugs (22,680 kg). Assume that the aerody namic data can be applied at fullscale. For level unaccelerated flight at V = 239 knots (123 m/s) of the fullscale aircraft, under the assumption that propulsion effects can be ignored,
(a) Find the limits on tail angle i, and CG position h imposed by the conditions C,,, > 0 and Cma < 0.
(b) For trimmed flight with 6, = 0, plot it vs. h for the aircraft and indicate where this line meets the boundaries of part (a).
Geometric Data
Wing area, S 1 .SO ft2 (0.139 m2)
Wing mean aerodynamic chord, ? 6.145 in (15.61 cm)
i, 15.29 in (38.84 cm) Tail area, S, 0.368 ft2 (0.0342 m2)
Aerodynamic Data
a ~b 0.077ldeg
a, 0.064ldeg
E , 0.72"
54 Chapter 2. Static Stability and ControlPart 1
Figure 2.29 Data for Exercise 2.6.
2.6" The McDonnell Douglas C17 is a fourengined jet STOL transport airplane.
(a) Find A and F for the wing using the geometrical data and Appendix C.
(b) Use Appendix B to estimate a,, the wing lift curve slope, assuming that P = 1 and K = 1.
(c) If a, = 0.068ldeg and a,, = a,, find the lift curve slope, a, of the aircraft. As 2aw
sume  =  (with a, expressed in rad'). aa TA
(d) Find Cma for the case where I , = 1, = 92 ft (28.04 m). Ignore propulsion effects.
*Problem courtesy of Professor E. K. Parks, University of Arizona.
Figure 2.30 Trim data for Exercise 2.6.
2.10 Exercises 55
(e) From the experimental curves of Figs. 2.29 and 2.30 and the given geometry, find Cmae and h,. Find CmU for h = 0.30.
Geometric Data
Wing area, S
Wing span, b
Root chord, c,
Tip chord, c,
chord line sweep, A
4 chord line sweep, A,,,
Tail area, St
2.7 Consider an aircraft with its tail identical to its wing (i.e., the same span, area, chord, etc.). Neglect body and wingbody interaction effects [i.e., in general ( IWb = ( ),I, neglect propulsion effects and assume zero elevator and tab deflections. As sume (2.2,7) in this instance is approximated by Mt = &, + M,,.,.
(a) What changes should be made in the expressions for a (2.3,18), Cmm (2.3,21a), and C,,,,, (2.3,22a)?
a€ (b) What would  have to be numerically in order that the neutral point h, lies aa
midway between the mean aerodynamic centers of the wing and tail?
(c) For trimmed level flight, derive an expression for the ratio of the lift generated by the wing to the lift generated by the tail as a function of the tail angle it. Assume
2.8" The following data were taken from a flight test of a PA32R300 Cherokee6 air plane.
Altitude VE Mass 1, XCG
Ift) (km) (mph) (m.1~) (slugs) ( k g ) (deg) (in) (cm)
4540 1.384 91.0 40.7 113.4 1656 1.5 93.89 238.5 4560 1.390 109 48.7 113.0 1650 0 93.89 238.5 4700 1.433 126 56.3 112.9 1649 1.0 93.89 238.5 4580 1.396 155 69.3 112.7 1646 2.0 93.89 238.5 5320 1.622 89.0 39.8 100.4 1466 4.5 86.82 220.5 4620 1.408 105 46.9 100.2 1463 2.0 86.82 220.5 4740 1.445 123 55.0 100.0 1461 0.3 86.82 220.5 4900 1.494 151 67.5 99.84 1458  1.0 86.82 220.5 4880 1.487 87.0 38.9 88.5 1 1293 7.2 80.43 204.3 4820 1.469 103 46.0 88.35 1290 3.5 80.43 204.3 4880 1.487 122 54.5 88.20 1288 1.5 80.43 204.3 4740 1.445 152 68.0 88.04 1286 0 80.43 204.3
*Problem courtesy of Professor E. K. Parks, University of Arizona.
56 Chapter 2. Static Stability and ControlPart I
The data were taken in trimmed level flight. x,, is the distance of the CG aft of the nose of the aircraft. The aircraft has an allmoving tail and thus it is used instead of 6, to trim the aircraft. The wing area is S = 174.5 ft2 (16.21 m2).
(a) Plot tailsetting angle, it, versus the lift coefficient of the aircraft for each of the three CG locations.
(b) Curve fit the data points in (a) with three straight lines having a common inter cept (refer to Fig. 2.18).
(c) Use a graphical technique to find the location of the neutral point (controls fixed) relative to the nose of the aircraft (refer to Fig. 2.21).
2.9 Starting with (2.6,l lb), derive (2.6,13).
2.10 The elevator control force to trim a particular airplane at a speed of 300 kts (154 d s ) is zero. Using the following data estimate the force required to change the trim speed to 310 kts (159 d s ) . Assume that C,, is sufficiently small that C,, = 0 can be used in the expression for control force.
Geometric Data
Elevator gearing, G 3"Iin (1.18"1cm)
Elevator area aft of hinge line, S, 40 ft2 (3.72 m2)
Mean elevator chord, Ce 2.0 ft (0.61 m)
VH 0.56 CG location, h
Wing loading, w
Aerodynamic Data
Elevator hinge moment coefficient,
0.38
50 psf (2,395 Pa)
a, 0.025ldeg Neutral point, elevator free, hk 0.45
2.11 A fatal airplane accident has led to a civil court case. It is alleged that the airplane in question was unstable"that the neutral point was ahead of the center of gravity."
You are called as an expert witness to explain to the court (i.e., to the judge and the attorneys) the meaning and importance of stability, center of gravity, and neutral point. You know that your audience is composed of intelligent persons educated in the humanities and law, but you must assume that they have only a rudimentary knowledge of science and mathematics.
Write an essay describing how you would meet this challenge. You are free to use di agrams and simple models, but you must avoid any use of mathematics. You should keep your language simple, avoiding any technical jargon. Even the word moment should be avoided. Your goal is to clarify, not to impress the court with your superior technical knowledge.
2.11 Additional Symbols Introduced in Chapter 2 57
2.11 Additional Symbols Introduced in Chapter 2
airplane liftcurve slope, aC,/aa, elevator fixed
airplane liftcurve slope, aCL/aa, elevator free
ac,/a s, wing liftcurve slope, aC,,"lda
wingbody liftcurve slope, aCLw,/aa
tail liftcurve slope, aC,/aa,
see Eq. 2.5,l
JChJJffI
achefa42 achefa sf length of mean aerodynamic chord
mean elevator chord (see Sec. 2.5)
D I ~ ~ V ~ S
elevator hingemoment coefficient, H J ~ ~ v ~ S , ~ ~
Ll$pv2S
~ , d 4 p v ~ s
tail lift coefficient, LjipV2S,
acja s, acjaQ M I ~ ~ V ~ S C
M , , . ~ / & ~ V ~ S Z
M,, , ,~~~V*SC
airplane pitchingmoment coefficient at zero a  cm,, airplane pitchingmoment coefficient at zero a,,
det
pitchingmoment coefficient of the propulsion units
M)4pV2SC
acmiaQ acm/aa acm/ase TI$pV2S
see (2.4,14)
wing drag
drag of the tail
freeelevator factor ( I  a,b,la&,) for aircraft with tails
G elevator gearing
H, elevator hinge moment
58 Chapter 2. Static Stability and ControlPart 1
h CG position, fraction of mean chord (see Fig. 2.9)
hn neutral point of airplane, fraction of mean chord, elevator fixed
h: neutral point of airplane, fraction of mean chord, elevator free
hnw neutral point of wing, fraction of mean chord
hnwb neutral point of the wingbody combination
hs static stability limit
1 , tailsetting angle (see Fig. 2.1 1)
Kn static margin, see (2.3,6) L airplane lift
Lw wing lift
Lwb lift of wingbody combination
Lt lift of the tail
1, distance between CG and tail mean aerodynamic center
see Fig. 2.12
M " Mach number
M pitching moment about the CG
Mw pitching moment of the wing about the CG
Macw pitching moment of the wing about its mean aerodynamic center
Macwb pitchi~g moment of the wingbody combination about its mean aerodynamic center
Mwb pitching moment, about the CG, of the wingbody combination
Mt pitching moment of tail about CG
P control force, positive to the rear
S wing area
Se span of elevator
Se area of elevator aft of hinge line
st area of tail T thrust
V true airspeed
Q VIV,
Ve reference equilibrium airspeed
VE equivalent airspeed (EAS), ~a VH horizontal tail volume, S,l,/SZ  VH s,ijs? W aircraft weight
w wing loading (W/S)
(Y angle of attack of the zero lift line of the airplane (elevator angle zero)
f f w
f f w b
f f t
6,
6, E
€0
P
Po
2.11 Additional Symbols Introduced in Chapter 2 59
angle of attack of the zero lift line of the wing
angle of attack of the zero lift line of the wingbody combination
angle of attack of the tail
elevator angle
tab angle
downwash angle
downwash when a,, = 0
air density
standard sealevel value of p (see Appendix D)
C H A P T E R 3
Static Stability and Control Part 2
3.1 ManeuverabilityElevator Angle per g
In this and the following sections, we investigate the elevator angle and control force required to hold the airplane in a steady pullup with load factor' n (Fig. 3.1). The concepts discussed here were introduced by S. B. Gates (1942). The flightpath tan gent is horizontal at the point under analysis, and hence the net normal force is L  W = (n  l)W vertically upward. The normal acceleration is therefore (n  1)g.
When the airplane is in straight horizontal flight at the same speed and altitude, the elevator angle and control force to trim are 8, and P, respectively. When in the pullup, these are changed to 6, + As, and P + AP. The ratios A8J(n  1) and APl(n  1) are known, respectively, as the elevator angle per g, and the control force per g. These two quantities provide a measure of the maneuverability of the airplane; the smaller they are, the more maneuverable it is.
The angular velocity of the airplane is fixed by the speed and normal accelera tion (Fig. 3.1).
As a consequence of this angular velocity, the field of the relative air flow past the airplane is curved. It is as though the aircraft were attached to the end of a whirling arm pivoted at 0 (Fig. 3.1). This curvature of the flow field alters the pressure distri bution and the aerodynamic forces from their values in translational flight. The change is large enough that it must be taken into account in the equations describing the motion.
We assume that q and the increments Aa, A6, etc. between the rectilinear and curved flight conditions are small, so that the increments in lift and moment may be written
where, in order to maintain a nondimensional form of equations, we have introduced the dimensionless pitch rate 4 = qc/2V, and CLq = aCL/a4, Cmq = aCm/a4. The q de
'The load factor is the ratio of lift to weight, n = L/W. It is unity in straight horizontal flight.
3.1 ManeuverabilityElevator Angle per g 61
$.w Figure 3.1 Airplane in a pullup.
rivatives are discussed in Sec. 5.4. In this form, these equations apply to any configu ration. From (3.1,l) we get
which is more conveniently expressed in terms of the weight coeficient Cw and the mass ratio /I. (see Sec. 3.15), that is,
Since the curved flight condition is also assumed to be steady, that is, without angular acceleration, then AC, = 0. Finally, we can relate AC, to n thus:
Equations (3.1,2 and 3) therefore become
62 Chapter 3. Static Stability and ControlPart 2
which are readily solved for Aa and Ase to yield the elevator angle per g
Aae   cw 1   [cma  n  1 det 2P (CLyCma  cLacrny)] (a)
(3.1.6)
and
where det is the same expression previously given in (2.4,13d). As has been shown in Sec. 2.4 det does not depend on CG position, hence the variation of A6Jn  1) with h is provided by the terms in the numerator. Writing Cmn = CLa(h  h,) (3.1,6a) be comes
The derivatives CLq and Cmq both in general vary with h, the former linearly, the latter quadratically, (see Sec. 5.4). Thus (3.1,7), although it appears to be linear in h, is not exactly so. For airplanes with tails, CLq can usually be neglected altogether when compared with 2p, and the variation of Cmy with h is slight. The equation is then very nearly linear with h, as illustrated in Fig. 3.2. For tailless airplanes, the variation may show more curvature. The point where A6J(n  1 ) is zero is called the controlfixed maneuver point, and is denoted by h,, as shown. From (3.1,7) we see that
where Cmq(hm) and CLy(hm) are the values of these two derivatives evaluated for h =
h,. When Cmq and CLq can be assumed to be independent of h, (3.1,7) reduces to
The difference (h,  h) is known as the controlfixed maneuver margin.
Controlfixed neutral point \h,
CG &sition, h
maneuver point
Figure 3.2 Elevator angle per g.
3.2 Control Force per g 63
3.2 Control Force per g
From (2.8,4) we get the incremental control force
AP = G S , C ~ $ ~ V ~ AC,, (3.2,l)
C,, is given for rectilinear flight by (2.5,2). Since it too will in general be influenced by q, we write for the incremental value (AS, = 0)
Ache = C , , ~ ( r + Cheq4 + b2ASe ( 3 . 2 2
The derivative C,,,, is discussed in Sec. 5.4. Using (3.1,4) and (3.1,6b), (3.2,2) is readily expanded to give
From (2.6,4b) we note that the last parenthetical factor is b2CijCL, or b2a11a. For A6, we use the approximation (3.1,9) in the interest of simplicity and the result for AC,, after some algebraic reduction is
where
Ache  CW arb,   n  1 2 p det ( 2 ~  CLy>(h  hk)
In keeping with earlier nomenclature, hk is the controlfree maneuver point and (hk  h) is the corresponding margin. On noting that CWipv2 is the wing loading w, we find the control force per g is given by
Note that this result applies to both tailed and tailless aircraft provided that the appro priate derivatives are used. The following conclusions may be drawn from (3.2,6).
1. The control force per g increases linearly from zero as the CG is moved for ward from the controlfree maneuver point, and reverses sign for h > hk.
2. It is directly proportional to the wing loading. High wing loading produces "heavier" controls.
3. For similar aircraft of different size but equal wing loading, Q SeZe; i.e. to the cube of the linear size.
4. Neither C, nor V enters the expression for Q explicitly. Thus, apart from M and Reynolds number effects, Q is independent of speed.
5. The factor p which appears in (3.25) causes the separation of the controlfree neutral and maneuver points to vary with altitude, size, and wing loading, in the same manner as the interval (h,  h,).
Figure 3.3 shows a typical variation of Q with CG position. The statement made above that the control force per g is "reversed" when h > h:, must be interpreted cor
64 Chapter 3. Static Stability and ControlPart 2
h CG position
Controlfree neutral point 1 Controlfree point of zero gradient of \ rnsneuver point
control force at point of zero control handsoff speed force per g
Figure 3.3 Control force per g.
rectly. In the first place this does not necessarily mean a reversal of control move ment per g , for this is governed by the elevator angle per g . If hk < h < h,, then there would be reversal of Q without reversal of control movement. In the second place, the analysis given applies only to the steady state at load factor n, and throws no light whatsoever on the transition between unaccelerated flight and the pullup condition. No matter what the value of h, the initial control force and movement re quired to start the maneuver will be in the normal direction (backward for a pullup), although one or both of them may have to be reversed before the final steady state is reached.
CONTROLFORCE GADGETS
The control forces on a manually controlled airplane can be made to deviate from the "natural" pattern that flows from the size of the airplane, the aerodynamic design, and the speed and altitude of flight without necessarily using either powered controls or aerodynamic tabs (see Sec. 2.7). Some "gadgets" that can be used to this end are springs, weights, and variableratio sprockets and linkages. These can have the effect of modifying the controlforce to trim and the controlforce per g , giving the pilot the same feel as if the controlfree neutral and maneuver points were moved. Some de tails of these effects are given in Sec. 7.1 of Etkin (1972).
3.3 Influence of HighLift Devices on Trim and Pitch Stiffness
Conventional airplanes utilize a wide range of aerodynamic devices for increasing CLWdx. These include various forms of trailing edge elements (plain flaps, split flaps,
3.3 Influence of HighLift Devices on Trim and Pitch Stiffness 65
slotted flaps, etc.), leading edge elements (drooped nose, slats, slots, etc.) and purely fluid mechanical solutions such as boundary layer control by blowing. Each of these has its own characteristic effects on the lift and pitching moment curves, and it is not feasible to go into them in depth here. The specific changes that result from the "con figurationtype" devices, i.e. flaps, slots, etc., can always be incorporated by making the appropriate changes to ha,",, CmrrCwh and CLwh in (2.2,4) and following through the consequences. Consider for example the common case of partspan trailing edge flaps on a conventional tailed airplane. The main aerodynamic effects of such flaps are illustrated in Fig. 3.4.'
1. Their deflection distorts the shape of the spanwise distribution of lift on the wing, increasing the vorticity behind the flap tips, as in (a).
2. They have the same effect locally as an increase in the wingsection camber, that is, a negative increment in C,,,"' and a positive increment in CLwb.
3. The downwash at the tail is increased; both E, and aelaa will in general change.
The change in wingbody C,,, is obtained from (2.2,4) as
ACmM,, = ACrn c,cw,, + A C L ~ ~ , J ~  hnM,J (3.3,l)
The change in airplane C, is
and the change in tail pitching moment is
When the increments ACmUcwb and LCLw, are constant with a and Ah,,,, is negligible, then the only effect on CLn and C,* is that of adaa, and from (2.3,18) and (2.3,21a) these are
The net result on the C, and C,,, curves is obviously very much configuration depen dent. If the Cm  a relation were as in Fig. 3.4e, then the trim change would be very large, from a , at Sf = 0 to a, after flap deflection. The C, at a, is much larger than at a , and hence if the flap operation is to take place without change of trim speed, a downelevator deflection would be needed to reduce atrim to a, (Fig. 3 . 4 ~ ) . This would result in a nosedown rotation of the aircraft.
'Note that a is still the angle of attack of the zerolift line of the basic configuration, and that the lift with flap deflected is not zero at zero a.
66 Chapter 3. Static Stability and ControlPart 2
Spanwise loadingflaps down
 OaPs Vdcity In wake /behind flap tips
i  L
I.  .
Zero lift line, 61 = 0 Mean aerodynamic center
A€
(4 Figure 3.4 Effect of partspan flaps. (a) Change of lift distribution and vorticity. (b) Changes in forces and moments. (c) Change in C,. (d) Change in downwash. (e) Change in C,.
3.4 Influence of the Propulsive System on Trim and Pitch Stiffness
The influences of the propulsive system upon trim and stability may be both impor tant and complex. The range of conditions to be considered in this connection is ex tremely wide. There are several types of propulsive units in common userecipro
3.4 Influence of the Propulsive System on Trim and Pitch Stiffness 67
catingenginedriven propellers, turbojets, turboprops, and rockets, and the variations in engineplusvehicle geometry are very great. The analyst may have to deal with such widely divergent cases as a highaspectratio straightwinged airplane with six wingmounted counterrotating propellers or a lowaspectratio delta with buried jet engines. Owing to its complexity, a comprehensive treatment of propulsive system influences on stability is not feasible. There does not exist sufficient theoretical or empirical information to enable reliable predictions to be made under all the above mentioned conditions. However, certain of the major effects of propellers and propul sive jets are sufficiently well understood to make it worth while to discuss them, and this is done in the following.
In a purely formal sense, of course, it is only necessary to add the appropriate di rect effects, Cmop and aCm;aa in (2.3,21 and 22), together with the indirect effects on the various wingbody and tail coefficients in order to calculate all the results with power on.
When calculating the trim curves (i.e., elevator angle, tab angle, and control force to trim) the thrust must be that required to maintain equilibrium at the condition of speed and angle of climb being investigated (see Sec. 2.4). For example (see Fig. 2. I), assuming that a, 4 1
C, = C, + C, sin y (a)
C, cos y = C, + C,a, (b) (3.4,l)
Solving for C , we get
C, + CL tan y C, =
1  a,tan y (3.432)
Except for very steep climb angles, a, tan y 4 1, and we may write approximately,
C, = C, + C, tan y (3.4,3)
Let the thrust line be offset by a distance z, from the CG (as in Fig. 3.5) and neglect ing for the moment all other thrust contributions to the pitching moment except Tz,,, we have
ZP Cm,, = CT 1
C
ZP = (C, + CL tan y) y
C
<\I rxpT disk A~rplane CG 4  zp_
q+ Thrust line (re\at\ve
7 w~nd)
Figure 3.5 Forces on a propeller.
68 Chapter 3. Static Stability and ControlPart 2
Now let C, be given by the parabolic polar (2.1,2), so that
CmP = (CDm,,, + KC,' + CL tan y) " C
(3.495)
Strictly speaking, the values of CD and C, in (3.4,4 and 5 ) are those for trimmed flight, i.e. with 6, = 6e,r,m. For the purposes of this discussion of propulsion effects we shall neglect the effects of 6, on CD and C,, and assume that the values in (3.43) are those corresponding to 6, = 0. The addition of this propulsive effect to the Cm curve for rectilinear gliding flight in the absence of aeroelastic and compressibility effects might then appear as in Fig. 3 . 6 ~ . We note that the gradient dCmldCL for any value of y > 0 is less than for unpowered flight. If dC,,,ldCL is used uncritically as a criterion for stability an entirely erroneous conclusion may be drawn from such curves.
1. Within the assumptions made above, the thrust moment Tz, is independent of a , hence aCmJaa = 0 and there is no change in the N P from that for unpow ered flight.
2. A true analysis of stability when both speed and a are changing requires that the propulsive system controls (e.g., the throttle) be keptfixed, whereas each point on the curves of Fig. 3 . 6 ~ corresponds to a different throttle setting. This parallels exactly the argument of Sec. 2.4 concerning the elevator trim slope.
Climbing flight Y > O * /
*
TI const
* p = const T0,
gliding flight
(b) Figure 3.6 Effect of direct thrust moment on C,(a) curves (see (2.41,d). (a) Constant y. (b) Constant thrust and power.
3.4 Influence of the Propulsive System on Trim and Pitch Stiffness 69
For in fact, under the stated conditions, the Cm  CI, curve is transformed into a curve of 6 e,r,m VS. V by using the relations 6 e,r,, = Cm(a)ICm, and C, =
WIip~2S. The slopes of Cm vs. C, and VS. V will vanish together.
If a graph of Cm vs. C, be prepared for fixed throttle, then y will be a variable along it, and its gradient dCmldC, is an index of stability, as shown in Sec. 6.4. The two idealized cases of constant thrust and constant power are of interest. If the thrust at fixed throttle does not change with speed, then we easily find
and (3.4,6)
If the power P is invariant, instead of the thrust, then T = P N and we find
whence
Thus in the constant thrust case, the poweroff Cm  C, graph simply has its slope changed by the addition of thrust, and in the constant power case the shape is changed as well. The form of these changes is illustrated in Fig. 3.6b and it is evident by comparison with 3 . 6 ~ that the behavior of dCmldC, is quite different in these two situations.
THE INFLUENCE OF RUNNING PROPELLERS
The forces on a single propeller are illustrated in Fig. 3.5, where ol, is the angle of at tack of the local flow at the propeller. It is most convenient to resolve the resultant into the two components T along the axis, and N,, in the plane of the propeller. The moment associated with T has already been treated above, and does not affect Cma. That due to N, is
where CNr, = N J ~ ~ V ~ S , , and Sp is the propeller disk area. To get the total ACm for sev eral propellers, increments such as (3.4,7) must be calculated for each and summed. Theory shows (Ribner, 1945) that for small angles CN, is proportional to ol,. Hence N,, contributes to both Cm,n and aCmr/aa. The latter is
acmr,  s, x,, a c N , acu,   aa s i: a% aa (3.43)
If the propeller were situated far from the flow field of the wing, then aaJaa would be unity. However, for the common case of wingmounted tractor propellers with the
70 Chapter 3. Static Stability and ControlPart 2
propeller plane close to the wing, there is a strong upwash E, at the propeller. Thus
and
where the constant in (3.4,9a) is the angle of attack of the propeller axis relative to the airplane zerolift line. Finally,
Increase of Wing Lift
When a propeller is located ahead of a wing, the highvelocity slipstream causes a distortion of the lift distribution, and an increase in the total lift. This is a principal mechanism in obtaining high lift on socalled deflected slipstream STOL airplanes. For accurate results that allow for the details of wing and flap geometry powered model testing is needed. However, for some cases there are available theoretical re sults (Ellis, 1971; Kuhn, 1959; Priestly, 1953) suitable for estimates. Both theory and experiment show that the lift increment tends to be linear in a for constant C, and hence has the effect of increasing a,,, the liftcurve slope for the wingbody combi nation. From (2.3,23) this is seen to reduce the effect of the tail on the NP location, and can result in a decrease of pitch stiffness.
Effects on the Tail
The propeller slipstream can affect the tail principally in two ways. (1) Depend ing on how much if any of the tail lies in it, the effective values of a, and a, will expe rience some increase. (2) The downwash values E, and adaa may be appreciably al tered in any case. Methods of estimating these effects are at best uncertain, and poweredmodel testing is needed to get results with engineering precision for most new configurations. However, some empirical methods (Smelt and Davies, 1937; Millikan, 1940; Weil and Sleeman, 1949) are available that are suitable for some cases.
THE INFLUENCE OF JET ENGINES
The direct thrust moment of jet engines is treated as shown at the beginning of this section, the constantthrust idealization given in (3.4,6) often being adequate. In addi tion, however, there is a normal force on jet engines as well as on propellers.
Jet Normal Force The air that passes through a propulsive duct experiences, in general, changes in
both the direction and magnitude of its velocity. The change in magnitude is the prin cipal source of the thrust, and the direction change entails a force normal to the thrust
3.4 Influence of the Propulsive System on Trim and Pitch Stiffness 71
line. The magnitude and line of action of this force can be found from momentum considerations. Let the mass flow through the duct be m' and the velocity vectors at the inlet and outlet be Vi and V,. Application of the momentum principle then shows that the reaction on the airplane of the air flowing through the duct is
F = rnl(V,  Vi) + F'
where F' is the resultant of the pressure forces acting across the inlet and outlet areas. For the present purpose, F' may be neglected, since it is approximately in the direc tion of the thrust T. The component of F normal to the thrust line is then found as in Fig. 3.7. It acts through the intersection of Vi and V,. The magnitude is given by
N, = mlVi sin 6
or, for small angles,
N, = m1Vi6
In order to use this relation, both Vi and 6 are required. It is assumed that Vi has that direction which the flow would take in the absence of the engine; that is, 13 equals the angle of attack of the thrust line a, plus the upwash angle due to wing induction E,.
It is further assumed that the magnitude Vi is determined by the mass flow and inlet area; thus
where Ai is the inlet area, and pi the air density in the inlet. We then get for N, the ex pression
The corresponding pitchingmoment coefficient is
Engine air . . .  ,Tail pipe
Figure 3.7 Momentum change of engine air.
72 Chapter 3. Static Stability and ControlPart 2
Figure 3.8 Jetinduced inflow.
Since the pitching moment given by (3.4,14) varies with a at constant thrust, then there is a change in Cma given by
mr2 1 a€. ax. ACma =   X j l + " +o" aipi iprrs? [ ( aa ) a,]
The quantities mr and pi can be determined from the engine performance data, and for subsonic flow, ae,/aa is the same as the value aeP/aa used for propellers. ax,/aa can be calculated from the geometry.
Jet Induced Inflow A spreading jet entrains the air that surrounds it, as illustrated in Fig. 3.8, thereby
inducing a flow toward the jet axis. If a tailplane is placed in the induced flow field, the angle of attack will be modified by this inflow. A theory of this phenomenon which allows for the curvature of the jet due to angle of attack has been formulated by Ribner (1946). This inflow at the tail may vary with a sufficiently to reduce the stability by a significant amount.
3.5 Effect of Structural Flexibility
Many vehicles when flying near their maximum speed are subject to important aero elastic phenomena. Broadly speaking, we may define these as the feedback effects upon the aerodynamic forces of changes in the shape of the airframe caused by the aerodynamic forces. No real structure is ideally rigid, and aircraft are no exception. Indeed the structures of flight vehicles are very flexible when compared with bridges, buildings, and earthbound machines. This flexibility is an inevitable characteristic of structures designed to be as light as possible. The aeroelastic phenomena which result may be subdivided under the headings static and dynamic. The static cases are those in which we have steadystate distortions associated with steady loads. Examples are aileron reversal, wing divergence, and the reduction of longitudinal stability. Dy namic cases include buffeting and flutter. In these the time dependence is an essential element. From the practical design point of view, the elastic behavior of the airplane affects all three of its basic characteristics: namely performance, stability, and struc
3.5 Effect of Structural Flexibility 73
tural integrity. This subject occupies a wellestablished position as a separate branch of aeronautical engineering. For further information the reader is referred to one of the books devoted to it (Bisplinghoff, 1962; Dowell, 1994).
In this section we take up by way of example a relatively simple aeroelastic ef fect; namely, the influence of fuselage flexibility on longitudinal stiffness and con trol. Assume that the tail load L, bends the fuselage so that the tail rotates through the angle Aa, = kL, (Fig. 3.9) while the wing angle of attack remains unaltered. The net angle of attack of the tail will then be
and the tail lift coefficient at 6, = 0 will be
CL, = atat = at(aWb  E  it  kLr)
But L, = CL>pV2~,, from which
Solving for C,,, we get
Comparison of (3.5,2) with (2.3,13) shows that the tail effectiveness has been re duced by the factor 141 + ka,(p/2)~2S,]. The main variable in this expression is V, and it is seen that the reduction is greatest at high speeds. From (2.3,23) we find that the reduction in tail effectiveness causes the neutral point to move forward. The shift is given by
Aa,  Ah, =  V, 1  
a ( where
1 Aa, = a,
1 + ka,4p~2S,
The elevator effectiveness is also reduced by the bending of the fuselage. For, if we consider the case when 6, is different from zero, then (3.5,l) becomes
Figure 3.9 Tail rotation due to fuselage bending.
74 Chapter 3. Static Stability and ControlPart 2
and (3.5,2) becomes
Thus the same factor 1/(1 + ka,p/2V2S,) that operates on the tail lift slope a, also mul tiplies the elevator effectiveness a,.
3.6 Ground Effect
At landing and takeoff airplanes fly for very brief (but none the less extremely impor tant) time intervals close to the ground. The presence of the ground modifies the flow past the airplane significantly, so that large changes can take place in the trim and sta bility. For conventional airplanes, the takeoff and landing cases provide some of the governing design criteria.
The presence of the ground imposes a boundary condition that inhibits the down ward flow of air normally associated with the lifting action of the wing and tail. The reduced downwash has three main effects, being in the usual order of importance:
1. A reduction in E, the downwash angle at the tail.
2. An increase in the wingbody lift slope a,,. 3. An increase in the tail lift slope a,.
The problem of calculating the stability and control near the ground then resolves it self into estimating these three effects. When appropriate values of ad&, awb, and a, have been found, their use in the equations of the foregoing sections will readily yield the required information. The most important items to be determined are the el evator angle and control force required to maintain CLax in level flight close to the ground. It will usually be found that the ratio a,la is decreased by the presence of the ground. (2.3,23) shows that this would tend to move the neutral point forward. How ever, the reduction in adaa is usually so great that the net effect is a large rearward shift of the neutral point. Since the value of C,, is only slightly affected, it turns out that the elevator angle required to trim at CLmaX is much larger than in flight remote from the ground. It commonly happens that this is the critical design condition on the elevator, and it will govern the ratio SJS,, or the forward CG limit (see Sec. 3.7).
3.7 CG Limits
One of the dominant parameters of longitudinal stability and control has been shown in the foregoing sections to be the foreandaft location of the CG (see Figs. 2.14, 2.18,2.19,2.25,2.27,2.28, and 3.2). The question now arises as to what range of CG position is consistent with satisfactory handling qualities. This is a critical design problem, and one of the most important aims of stability and control analysis is to provide the answer to it. Since aircraft always carry some disposable load (e.g., fuel, armaments), and since they are not always loaded identically to begin with (varia tions in passenger and cargo load), it is always necessary to cater for a variation in the CG position. The range to be provided for is kept to a minimum by proper loca
3.7 CG Limits 75
tion of the items of variable load, but still it often becomes a difficult matter to keep the handling qualities acceptable over the whole CG range. Sometimes the problem is not solved, and the airplane must be subjected to restrictions on the foreandaft dis tribution of its variable load when operating at part load.
THE AFT LIMIT
The most rearward allowable location of the CG is determined by considerations of longitudinal stability and control sensitivity. The behavior of the five principal con trol gradients are summarized in Fig. 3.10 for the case when the aerodynamic coeffi cients are independent of speed. From the handling qualities point of view, none of the gradients should be "reversed," that is, they should have the signs associated with low values of h. When the controls are reversible, this requires that h < h:. If the controls are irreversible, and if the artificial feel system is suitably designed, then the control force gradient aPIaV can be kept negative to values of h > h:, and the rear limit can be somewhat farther back than with reversible controls. The magnitudes of the gradients are also important. If they are allowed to fall to very small values the vehicle will be too sensitive to the controls. When the coefficients do not depend on speed, as assumed for Fig. 3.10, the NP also gives the stability boundary (this is proved in Chap. 6), the vehicle becoming unstable for h > h: with free controls or h > h, with fixed controls. If the coefficients dependent on speed, for example, C, = C,(M), then the CG boundary for stability will be different and may be forward of the NP.
Figure 3.10 The five control gradients.
76 Chapter 3. Static Stability and ControCPart 2
As noted in Chap. 1, it is possible to increase the inherent stability of a flight ve hicle. Stability augmentation systems (SAS) are in widespread use on a variety of air planes and rotorcraft. If such a system is added to the longitudinal controls of an air plane, it permits the use of more rearward CG positions than otherwise, but the risk of failure must be reckoned with, for then the airplane is reduced to its "inherent" sta bility, and would still need to be manageable by a human pilot.
THE FORWARD LIMIT
As the CG moves forward, the stability of the airplane increases, and larger control movements and forces are required to maneuver or change the trim. The forward CG limit is therefore based on control considerations and may be determined by any one of the following requirements:
1. The control force per g shall not exceed a specified value.
2. The controlforce gradient at trim, aPIaV, shall not exceed a specified value.
3. The control force required to land, from trim at the approach speed, shall not exceed a specified value.
4. The elevator angle required to land shall not exceed maximum up elevator.
5. The elevator angle required to raise the nosewheel off the ground at takeoff speed shall not exceed the maximum up elevator.
3.8 Lateral Aerodynamics
In the preceding sections of this and the previous chapter we discussed aerodynamic characteristics of symmetrical configurations flying with the velocity vector in the plane of symmetry. As a result the only nonzero motion variables were V, a, and q, and the only nonzero forces and moments were T, D, L, and M. We now turn to the cases in which the velocity vector is not in the plane of symmetry, and in which yaw ing and rolling displacements (P, +) are present. The associated force and moment coefficients are C,, C,, and C,.
One of the simplifying aspects of the longitudinal motion is that the rotation is about one axis only (the y axis), and hence the rotational stiffness about that axis is a very important criterion for the dynamic behavior. This simplicity is lost when we go to the lateral motions, for then the rotation takes place about two axes (x and z). The moments associated with these rotations are crosscoupled, that is, roll rotation p pro duces a yawing moment C, as well as rolling moment C,, and yaw displacement p and rate r both produce rolling and yawing moments. Furthermore, the roll and yaw controls are also often crosscoupleddeflection of the ailerons can produce signifi cant yawing moments, and deflection of the rudder can produce significant rolling moments.
Another important difference between the two cases is that in "normal" flight that is, steady rectilinear symmetric motion, all the lateral motion and force variables are zero. Hence there is no fundamental trimming problemthe ailerons and rudder would be nominally undeflected. In actuality of course, these controls do have a sec ondary trimming function whenever the vehicle has either geometric or inertial asym metriesfor example, one engine off, or multiple propellers all rotating the same
3.9 Weathercock Stability (Yaw Stiffness) 77
way. Because the gravity vector in normal flight also lies in the plane of symmetry, the CG position is not a dominant parameter for the lateral characteristics as it is for the longitudinal. Thus the CG limits, (see Sec. 3.7) are governed by considerations deriving from the longitudinal characteristics.
3.9 Weathercock Stability (Yaw Stiflness)
Application of the static stability principle to rotation about the z axis suggests that a stable airplane should have "weathercock stability. That is, when the airplane is at an angle of sideslip relative to its flight path (see Fig. 3.1 I), the yawing moment produced should be such as to tend to restore it to symmetric flight. The yawing mo ment N is positive as shown. Hence the requirement for yaw stiffness is that aNIaP must be positive. The nondimensional coefficient of N is
and hence for positive yaw stiffness aC,lap must be positive. The usual notation for this derivative is
This quantity is analogous in some respects to the longitudinal stability parameter C,,,m. It is estimated in a similar way by synthesis of the contributions of the various components of the airplane. The principal contributions are those of the body and the verticaltail surface. By contrast with Cma, the wing has little influence in most cases, and the CG location is a weak parameter. Whether or not a positive value of C,, will produce lateral stability can only be determined by a full dynamic analysis such as is done in Chap. 6.
Figure 3.11 Sideslip angle and yawing moment.
78 Chapter 3. Static Stability and ControlPart 2
Some data for estimating the contribution of the body to C,, is contained in Ap pendix B. There are also given data suitable for estimation of the liftcurve slope of the verticaltail surface. This may be used to calculate the tail contribution as shown below.
In Fig. 3.12 are shown the relevant geometry and the lift force LF acting on the vertical tail surface. If the surface were alone in an airstream, the velocity vector V, would be that of the free stream, so that (cf. Fig. 3.1 1) a, would be equal to P. When installed on an airplane, however, changes in both magnitude and direction of the local flow at the tail take place. These changes may be caused by the propeller slipstream, and by the wing and fuselage when the airplane is yawed. The angular de flection is allowed for by introducing the sidewash angle u, analogous to the down wash angle E. u is positive when it corresponds to a flow in the y direction; that is, when it tends to increase cu,. Thus the angle of attack is
and the lift coefficient of the verticaltail surface is
The lift is then
and the yawing moment is
Figure 3.12 Verticaltail sign conventions.
3.9 Weathercock Stability (Yaw Stiffness) 79
thus
The ratio SFIJSb is analogous to the horizontaltail volume ratio, and is therefore called the verticaltail volume ratio, denoted here by V,. Equation 3.9,5 then reads
and the corresponding contribution to the weathercock stability is
Generally speaking, the sidewash is difficult to estimate with engineering precision. Suitable windtunnel tests are required for this purpose. The contribution from the fuselage arises through its behavior as a lifting body when yawed. Associated with the side force that develops is a vortex wake which induces a lateralflow field at the tail. The sidewash from the propeller is associated with the side force which acts upon it when yawed, and may be estimated by the method of (Ribner, 1944). The contribution from the wing is associated with the asymmetric structure of the flow which develops when the airplane is yawed. This phenomenon is especially pro nounced with lowaspectratio swept wings. It is illustrated in Fig. 3.13.
Figure 3.13 Vortex wake of yawed wing.
80 Chapter 3. Static Stability and ControlPart 2
THE VELOCITY RATIO V,/V
When the vertical tail is not in a propellor slipstream, V,/V is unity. When it is in a slipstream, the effective velocity increment may be dealt with as for a horizontal tail.
CONTRIBUTION OF PROPELLER NORMAL FORCE
The yawing moment produced by the normal force that acts on the yawed propeller is calculated in the same way as the pitchingmoment increment dealt with in Sec. 3.4. The result is similar to (3.4,8):
This is known as the propeller fin effect and is negative (i.e., destabilizing) when the propeller is forward of the CG, but is usually positive for pusher propellers. There is a similar yawing moment effect for jet engines (see Exercise 3.7).
3.10 Yaw Control
In most flight conditions it is desired to maintain the sideslip angle at zero. If the air plane has positive yaw stiffness, and is truly symmetrical, then it will tend to fly in this condition. However, yawing moments may act upon the airplane as a result of unsymmetrical thrust (e.g., one engine inoperative), slipstream rotation, or the un symmetrical flow field associated with turning flight. Under these circumstances, P can be kept zero only by the application of a control moment. The control that pro vides this is the rudder. Another condition requiring the use of the rudder is the steady sideslip, a maneuver sometimes used, particularly with light aircraft, to in crease the drag and hence the glide path angle. A major point of difference between the rudder and the elevator is that for the former trimming the airplane is a secondary and not a primary function. Apart from this difference, the treatment of the two con trols is similar. From (3.9,3) and (3.9,6), the rate of change of yawing moment with rudder deflection is given by
a cn 2 c = as, (3.10,l)
This derivative is sometimes called the "rudder power." It must be large enough to make it possible to maintain zero sideslip under the most extreme conditions of asymmetric thrust and turning flight.
A second useful index of the rudder control is the steady sideslip angle that can be maintained by a given rudder angle. The total yawing moment during steady sideslip may be written
Cn = CnpP + CnarSr
For steady motion, C, = 0, and hence the desired ratio is
3.11 Roll Stiffness 81
The rudder hinge moment and control force are also treated in a manner similar to that employed for the elevator. Let the rudder hingemoment coefficient be given
by
C,, = b,aF + b2Sr (3.10,4)
The rudder pedal force will then be given by
where G is the rudder system gearing. The effect of a free rudder on the directional stability is found by setting C,, = 0
in (3.10,4). Then the rudder floating angle is
The verticaltail lift coefficient with rudder free is found from (3.9,3) to be
The free control factor for the rudder is thus seen to be of the same form as that for the elevator (see Sec. 2.6) and to have a similar effect.
3.1 1 Roll Stiffness
Consider a vehicle constrained, as on bearings in a wind tunnel, to one degree of freedomrolling about the x axis. The forces and moments resulting from a fixed dis placement 4 are fundamentally different in character from those associated with the rotations a and p. In the first place if the x axis coincides with the velocity vector V, no aerodynamic change whatsoever follows from the fixed rotation 4 (see Fig. 3.14). The aerodynamic field remains symmetrical with respect to the plane of symmetry, the resultant aerodynamic force remains in that plane, and no changes occur in any of the aerodynamic coefficients. Thus the roll stiffness aC,/a+ = C,, is zero in that case.
If the x axis does not coincide with V, then a secondorder roll stiffness results through the medium of the derivative aC,lap = C,,. Let the angle of attack of the x axis be a, (see Fig. 1.7), then the velocity vector when 4 = 0 is
After rolling through angle 4 about Ox, the x component of the velocity vector re mains unchanged, but the component V sin ax has projections on both of the new y
82 Chapter 3. Static Stability and ControlPart 2
I ,t Lift
t I W (weight)
Figure 3.14 Rolled airplane.
and z axes. Thus there is now a sideslip, and hence, an angle P and a resulting rolling moment. Using the notation of Appendix A.4, we get for the velocity vector in the new reference frame after the rotation
V 2 = L l ( 4 ) V , =
V sin ax cos + Thus the sideslip component is v = V sin ax sin 4 , and the sideslip angle is
U /3 = sin'  = sin' (sin ax sin +) (3.1 1,3) v
As a result of this positive p, and the usually negative C,, there is a restoring rolling moment C,,P, that is,
AC, = C,, sin' (sin ax sin 4 ) (a)
For small ax, we get the approximate result
AC, = C,, sin' (ax sin +) = CI,ax sin + (b) (3.11,4)
and if 4 also is small,
ACI = CI,q+ (c)
The stiffness derivative for rolling about Ox is then from (3.11,4a)
ac,  sin a; cos +  a+ c'p (I  sin2 ax sin2 4)'"
or for ru,+ 1,
3.12 The Derivative C,, 83
or for ax, 4 <. 1
Thus there is a roll stiffness that resists rolling if a, is >0, and would tend to keep the wings level. If rolling occurs about the wind vector, the stiffness is zero and the vehi cle has no preferred roll angle. If cu, < 0, then the stiffness is negative and the vehicle would roll to the position 4 = 1 80°, at which point C, = 0 and C,, < 0.
The above discussion applies to a vehicle constrained, as stated, to one degree of freedom. It does not, by any means, give the full answer for an unconstrained air plane to the question: "What happens when the airplane rolls away from a wings level attitudedoes it tend to come back or not?'That answer can only be provided by a full dynamic analysis like the lund 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 x axis, not the velocity vector, and hence experience the roll stiffness effect at the beginning of the response.
Even though airplanes have no firstorder aerodynamic roll stiffness, stable air planes do have an inherent tendency to fly with wings level. They do so because of what is known as the dihedral effect. This is a complex pattern involving gravity and the derivative C,,, which owes its existence largely to the wing dihedral (see Sec. 3.12). When rolled to an angle 4, there is a weight component mg sin 4 in the y di rection (Fig. 3.14). This induces a sideslip velocity to the right, with consequent /3 > 0, and a rolling moment Clop that tends to bring the wings level. The rolling and yawing motions that result from such an initial condition are, however, strongly cou pled, so no significant conclusions can be drawn about the behavior except by a dy namic analysis (see Chap. 6).
3.12 The Derivative C,,
The derivative C,, is of paramount importance. We have already noted its relation to roll stiffness and to the tendency of airplanes to fly with wings level. The primary contribution to C,, is from the wingits dihedral angle, aspect ratio, and sweep all being important parameters.
The effect of wing dihedral is illustrated in Fig. 3.15. With the coordinate system shown, the normal velocity component V, on the right wing panel (R) is, for small di hedral angle T,
V, = w cos r + v sin r = w + v r
and that on the other panel is w  v r . The terms +vT/V = represent opposite changes in the angle of attack of the two panels resulting from sideslip. The "up wind" panel has its angle of attack and therefore its lift increased, and vice versa. The result is a rolling moment approximately linear in both /3 and r, and hence a fixed
J
Figure 3.15 Dihedral effect. V, = normal velocity of panel R = w cos r + v sin r = w + uT :. vr vpr
Aa of R due to dihedral =  =  = pr. v v
value of C,, for a given r. This part of C,, 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 C,, proportional to C,. Consider the case of Fig. 3.16. The vertical induced velocity (downwash) of the vortex wake is greater at L than at R simply by virtue of the geometry of the wake. Hence the local wing angle of attack and lift are less at L than at R, and a negative C, results. Since this effect depends, essentially linearly, on the strength of the vortex wake, which is itself proportional to the wing C,, then the result is LCI, C,.
wake
Figure 3.16 Yawed lifting wing.
3.12 The Derivative C,, 85
Figure 3,17 Influence of body on C,g.
INFLUENCE OF FUSELAGE 9 N Cl,
The flow field of the body interacts with the wing in such a way as to modify its di hedral effect. To illustrate this, consider a long cylindrical body, of circular cross sec tion, yawed with respect to the main stream. Consider only the crossflow component of the stream, of magnitude VP, and the flow pattern which it produces about the body. This is illustrated in Fig. 3.17. It is clearly seen that the body induces vertical velocities which, when combined with the mainstream velocity, alter the local angle of attack of the wing. When the wing is at the top of the body (highwing), then the angleofattack distribution is such as to produce a negative rolling moment; that is, the dihedral effect is enhanced. Conversely, when the airplane has a low wing, the di hedral effect is diminished by the fuselage interference. The magnitude of the effect is dependent upon the fuselage length ahead of the wing, its crosssection shape, and the planform and location of the wing. Generally, this explains why highwing air planes usually have less wing dihedral than lowwing airplanes.
INFLUENCE OF SWEEP ON C,,
Wing sweep is a major parameter affecting C,,. Consider the swept yawed wing illus trated in Fig. 3.18. According to simple sweep theory it is the velocity V, normal to a wing reference line (i chord line for subsonic, leading edge for supersonic) that de termines the lift. It follows obviously that the lift is greater on the right half of the
Figure 3.18 Dihedral effect of a swept wing.
86 Chapter 3. Static Stability and ControlPart 2
Mean
v+ CG
Figure 3.19 Dihedral effect of the vertical tail.
wing shown than on the left half, and hence that there is a negative rolling moment. The rolling moment would be expected for small P to be proportional to
C,[(v;>,,,,  (v;)leftl = CLv2[c0s2 (A  P)  COS' (A + PI1 = 2c,Pv2 sin 2A
The proportionality with C, and P is correct, but the sin 2A factor is not a good ap proximation to the variation with A. The result is a C,, = C,, that can be calculated by the methods of linear wing theory.
INFLUENCE OF FIN ON C,,
The sideslipping airplane gives rise to a side force on the vertical tail (see Sec. 3.9). When the mean aerodynamic center of the vertical surface is appreciably offset from the rolling axis (see Fig. 3.19) then this force may produce a significant rolling mo ment. We can calculate this moment from (3.9,3). When the rudder angle is zero, that is, 8, = 0, the moment is found to be
thus
and
3.13 Roll Control
The angle of bank of the airplane is controlled by the ailerons. The primary function of these controls is to produce a rolling moment, although they frequently introduce a yawing moment as well. The effectiveness of the ailerons in producing rolling and yawing moments is described by the two control derivatives aC,/a8, and aC,,/a8,. The aileron angle 6, is defined as the mean value of the angular displacements of the two ailerons. It is positive when the right aileron movement is downward (see Fig. 3.20). The derivative aC,/a8, is normally negative, right aileron down producing a roll to the left.
3.13 Roll Control 87
Figure 3.20 Aileron angle.
For simple flaptype ailerons, the increase in lift on the right side and the de crease on the left side produce a drag differential that gives a positive (noseright) yawing moment. Since the normal reason for moving the right aileron down is to ini tiate a turn to the left, then the yawing moment is seen to be in just the wrong direc tion. It is therefore called aileron adverse yaw. On highaspectratio airplanes this tendency may introduce decided difficulties in lateral control. Means for avoiding this particular difficulty include the use of spoilers and Frise ailerons. Spoilers are il lustrated in Fig. 3.21. They achieve the desired result by reducing the lift and increas ing the drag on the side where the spoiler is raised. Thus the rolling and yawing mo ments developed are mutually complementary with respect to turning. Frise ailerons diminish the adverse yaw or eliminate it entirely by increasing the drag on the side of the upgoing aileron. This is achieved by the shaping of the aileron nose and the choice of hinge location. When deflected upward, the gap between the control sur face and the wing is increased, and the relatively sharp nose protrudes into the stream. Both these geometrical factors produce a drag increase.
Section through spoiler
Figure 3.21 Spoilers.
88 Chapter 3. Static Stability and ControlPart 2
The deflection of the ailerons leads to still additional yawing moments once the airplane starts to roll. These are caused by the altered flow about the wing and tail. These effects are discussed in Sec. 5.7 (Cnp), and are illustrated in Figs. 5.12 and 5.15.
A final remark about aileron controls is in order. They are functionally distinct from the other two controls in that they are rate controls. If the airplane is restricted only to rotation about the x axis, then the application of a constant aileron angle re sults in a steady rate of roll. The elevator and rudder, on the other hand, are displace ment controls. When the airplane is constrained to the relevant single axis degree of freedom, a constant deflection of these controls produces a constant angular displace ment of the airplane. It appears that both rate and displacement controls are accept able to pilots.
AILERON REVERSAL
There is an important aeroelastic effect on roll control by ailerons that is significant on most conventional airplanes at both subsonic and supersonic speeds. This results from the elastic distortion of the wing structure associated with the aerodynamic load increment produced by the control. As illustrated in Fig. 2.23, the incremental load caused by deflecting a control flap at subsonic speeds has a centroid somewhere near the middle of the wing chord. At supersonic speeds the control load acts mainly on the deflected surface itself, and hence has its centroid even farther to the rear. If this load centroid is behind the elastic axis of the wing structure, then a nosedown twist of the main wing surface results. The reduction of angle of attack corresponding to 6, > 0 causes a reduction in lift on the surface as compared with the rigid case, and a consequent reduction in the control effectiveness. The form of the variation of C,, with ipv2 for example can be seen by considering an idealized model of the phenom enon. Let the aerodynamic torsional moment resulting from equal deflection of the two ailerons be T(y) = BpV26, and let T(y) be antisymmetric, T(y) = T(y). The twist distribution corresponding to T(y) is @), also antisymmetric, such that @) is proportional to T at any reference station, and hence proportional to ipV26,. Finally, the antisymmetric twist causes an antisymmetric increment in the lift distribution, giving a proportional rolling moment increment AC, = kipV26,. The total rolling mo ment due to aileron deflection is then
and the control effectiveness is
As noted above, (C,,),,, is negative, and k is positive if the centroid of the aileron induced lift is aft of the wing elastic axis, the common case. Hence c,,I diminishes with increasing speed, and vanishes at some speed VR, the aileron reversal speed. Hence
3.14 Exercises 89
Substitution of (3.13,3) into (3.13,2) yields
This result, of course, applies strictly only if the basic aerodynamics are not Mach number dependent, i.e. so long as VR is at a value of M appreciably below 1.0. Other wise k and (C,,),igid are both functions of M, and the equation corresponding to (3.13,4) is
where MR is the reversal Mach number. It is evident from (3.13,4) that the torsional stiffness of the wing has to be great
enough to keep VR appreciably higher than the maximum flight speed if unacceptable loss of aileron control is to be avoided.
3.14 Exercises
3.1 Derive (3.1,4).
3.2 Derive an expression for the elevator angle per g in dimensional form. Denote the de rivatives of L and M with respect to a and q by aL/aq = L,, and so on. There are two choices: (1) do the derivation in dimensional form from the beginning, or (2) convert the nondimensional result (3.1,6) to dimensional form. Do it both ways and check that they agree.
3.3 Calculate the variation of the control force per g with altitude from the following data. Ignore propulsion effects.
Geometric Data
Weight, W
Wing area, S
Wing mean aerodynamic chord, c it
Tail area, S,
S<, Mean elevator chord,
G
Aerodynamic Data
90 Chapter 3. Static Stability and ControlPart 2
3.4 A small manually controlled airplane has an undesirable handling characteristicthe 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.
3.5 The range of elevator motion on an airplane is from 20" down to 30" up. Use Table 1.3 as a guide, a fellow student as a model, and a tape measure to arrive at a reason able value for the elevator gearing ratio G.
3.6 Two airplanes are similar, but one is jetpropelled and the other has a piston engine and propeller. The thrust line in each case is well below the CG with z,l? = 0.4. The poweroff pitching moment at 6, = 0 is Cm = 0.1  0.2 C,. The throttle is set to give level flight with C, = 0.4 and U D = 12. Consider several steady rectilinear flight conditions having the same throttle setting but different elevator settings, C, values and flightpath angles. Find dCmldC, for the two airplanes when passing through the altitude corresponding to the level flight conditions. As indicated in Sec. 3.4 and (6.4,lO) dCJdC, is an index of static longitudinal stability under certain conditions. Assuming that these conditions are met in this problem, how will the static longitudi nal stability of the two aircraft change as the aircraft slow down?
ac, 3.7 Derive an expression for the increment A 
ap attributable to a jet engine. (Hint, re
fer to (3.4,15).)
3.8 Suppose that as a result of an accident in flight the rear fuselage of an airplane is damaged, so that the flexibility parameter k in (3.5,l) et seq. is suddenly increased. The effect is large enough that the pilot notices a loss in longitudinal stability and control. Bearing in mind that the integrity of the fuselage structure depends on the tail load L, and the stability and control on the factor in parentheses in (3.5,4), ana lyze how the situation changes as the pilot slows down and descends to an emergency landing. Consider two cases; (1) C,, initially positive, (2 ) C,, initially negative.
3.9 Derive (3.9,8). Explain clearly each step in the development and justify any assump tions you make.
3.10 Use Appendix B to determine the elevator hinge moment parameters 6 , and 6, for a NACA 0009 airfoil (a symmetric airfoil with a thicknesstochord ratio of t/c = 0.09). The elevator has an elliptic nose, a sealed gap and a balance ratio of 0.2. In us
ing the curves assume that transition is at the leading edge; R = lo7; tan (i) =
tan  = 0.12; F, = 1; c+c = 0.325; A = 4.84; M = 0.
3.15 Additional Symbols Introduced in Chapter 3 91
3.15 Additional Symbols Introduced in Chapter 3
ac,,/aff,
ac,/aff, span of airplane
rollingmoment coefficient, L14pV2Sb
acia p verticaltail lift coefficient
ac,/aij damping in pitch, aC,/aij
yawingmoment coefficient, Nl$pV2Sb
acniap ac,/as, T I ~ P V ~ S
witpv2s diameter of propeller or jet
acceleration due to gravity
maneuver point, stick fixed (see 3.1,8)
maneuver point, stick free (see 3.2,5)
rolling moment
verticaltail lift
see Fig. 3.12
airplane mass
a ~ ~ a ~ load factor, UW
yawing moment
angular velocity in pitch, radls
qFl2 v verticaltail area
propeller disk area
thrust of one propulsion unit
effective velocity vector at the fin
verticaltail volume ratio, S,lJSb
wing loading, W/S
see Fig. 3.19
effective angle of attack of the fin, (see Fig. 3.12)
sideslip angle
dihedral angle
92 Chapter 3. Static Stability and ControlPart 2
E,, upwash at propeller
6, aileron angle (see Fig. 3.20)
6, rudder angle (see Fig. 3.1 2 )
A sweepback angle a sidewash angle
p 2mlpSC
bank angle
C H A P T E R 4
General Equations of Unsteady Motion
4.1 General Remarks
The basis for analysis, computation, or simulation of the unsteady motions of a flight vehicle is the mathematical model of the vehicle and its subsystems. An airplane in flight is a very complicated dynamic system. It consists of an aggregate of elastic bodies so connected that both rigid and elastic relative motions can occur. For exam ple, the propeller or jetengine rotor rotates, the control surfaces move about their hinges, and bending and twisting of the various aerodynamic surfaces occur. The ex ternal forces that act on the airplane are also complicated functions of its shape and its motion. It seems clear that realistic analyses of engineering precision are not likely to be accomplished with a very simple mathematical model. The model that is developed in the following has been found by aeronautical engineers and researchers to be very useful in practise. We begin by first treating the vehicle as a single rigid body with six degrees of freedom. This body is free to move in the atmosphere under the actions of gravity and aerodynamic forcesit is primarily the nature and com plexity of aerodynamic forces that distinguish flight vehicles from other dynamic systems. Next we add the gyroscopic effects of spinning rotors and then continue with a discussion of structural distortion (aeroelastic effects).
As was noted in Chap. 1, the Earth is treated as flat and stationary in inertial space. These assumptions simplify the model enormously, and are quite acceptable for dealing with most problems of airplane flight. The effects of a round rotating Earth are treated at some length in Etkin (1972).
Extensive use is made in the developments that follow of linear algebra, with which the reader is assumed to be familiar. Appendix A.l contains a brief review of some pertinent material.
4.2 The RigidBody Equations
In the interest of completeness, the rigidbody equations are derived from first princi ples, that is to say, we apply Newton's laws to an element dm of the airplane, and then integrate over all elements. The velocities and accelerations must of course be relative to an inertial, or Newtonian, frame of reference. As we noted in Sec. 1.6 the frame F,, fixed to the Earth, is assumed to be such a frame. We also noted there that velocities relative to F, are identified by a superscript E. In order to avoid the carry ing of the cumbersome superscript throughout the following development, we shall
94 Chapter 4. General Equations of Unsteady Motion
temporarily assume that W = 0 in (1.6,1), so that VE = V, and make an appropriate adjustment at the end. In the frame FB
V, = [u v wIT ( 4 2 1 )
The position vector of dm relative to the origin of FE is r, + r (see Fig. 4.1). In the frame FE,
r ~ E = [xE YE zEIT (4.22)
and in the frame FB
r, = [X y zIT
The inertial velocity of dm is
vE = (r,, + rE) = VE + rE (4.2?4)
The momentum of dm is vdm, and of the whole airplane is
Figure 4.1 Axes.
4.2 The RigidBody Equations 95
Since C is the mass center, the last integral in (4.2,5) is zero and
where m is the total mass of the airplane. Newton's second law applied to dm is
d f , = v,dm (4.2,7)
where d f , is the resultant of all forces acting on dm. The integral of (4.2,7) is
or, from (4.2,6)
fE = m v E (4.2,8)
The quantity Jdf , is a summation of all the forces that act upon all the elements. The internal forces, that is, those exerted by one element upon another, all occur in equal and opposite pairs, by Newton's third law, and hence contribute nothing to the summation. Thus f , is the resultant external force acting upon the airplane.
This equation relates the external force on the airplane to the motion of the CG. We need also the relation between the external moment and the rotation of the air 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 dh = r X vdm. It is convenient in the following to use the matrix form of the cross product (see Appendix A.l ) so that
dh, = f,v,dm
Consider
Now from (4.2,4), 
f E = OE  V E
and the moment of d f about C is
dG = r X d f
so that from (4.2,7)
dG, = fEdfE = f,v,dm
Equation (4.2,9) then becomes
d dG, =  (dh,)  (O ,  v,)v,dm
dt
Since v x v = 0, ( 4 . 2 , l l ) becomes
d dGE =  (dh,) + V,v,dm
dt
96 Chapter 4. General Equations of Unsteady Motion
Equation (4.2,12) is now integrated as was (4.2,7), and becomes
By an argument similar to that for Jdf , JdG is shown to be the resultant external moment about C, denoted G. Jdh is called the moment of momentum, or angular mo mentum of the airplane and is denoted h. Formulas for h are derived in Sec. 4.3. Us ing (4.2,6) and noting that V X V = 0, (4.2,13) reduces to
where
The reader should note that, in (4.2,14), both G and h are referred to a moving point, the mass center. For a moving reference point other than the mass center, the equation does not in general apply. The reader should also note that (4.2,8) and (4.2,14) are both valid when there is relative motion of parts of the airplane.
The two vector equations of motion of the airplane, equivalent to six scalar equa tions, are (4.2,8) and (4.2,14)
When the wind vector W is not zero, the velocity V, in (4.2,5) is that of the CG rela tive to F,. The angular momentum h is the same whether W is zero or not (see Exer cise 4.1). Hence in the general case when wind is present the equations of motion are:
4.3 Evaluation of the Angular Momentum h
We shall want the angular momentum components in FB. Now
h = ! d h = l r x v d m
So in FB we have
where
4.3 Evaluation of the Angular Momentum h 97
Let the angular velocity of the airplane relative to inertial space' be
= [ P q rlT where p, q, r are the rates of roll, pitch and yaw respectively (see Fig. 1.6).
Now the velocity of a point in a rigid rotating body is given by2
v, = V , + O,r, (a) where ( 4 . 3 3
so that
r (V, + &,r,)dm hB = I .
= (1 iB dm) vB + 1 iBhBrB dm (4.3,3)
The first integral in (4.3,3) vanishes since the origin of r is the CG. When the triple matrix product of the second integral is expanded (see Exercise 4.2) we get the result for h,:
where
and
IX = I @ + ?)dm; I, = (x2 + z2)dm; I, = 1 (x2 + y2)dm
I I ( 4 . W
IXY IYX = X Y dm; I,, = I, = xz dm; I,, = I,, = I yz dm
I, is the inertia matrix, its elements being the moments and products of inertia of the airplane. When the xz plane is a plane of symmetry, which is the usual assumption, then
Iv = Iyz = 0
and the only offdiagonal term remaining is I,. If the direction of the x axis is so cho sen that this product of inertia also vanishes, which is always possible in principle, then the axes are principal axes.
'Since there is no need to use the angular velocity relative to any other frame of reference the distin guishing superscript E is not needed on w.
'See Appendix A.6
98 Chapter 4. General Equations of Unsteady Motion
4.4 Orientation and Position of the Airplane
The position and orientation of the airplane are given relative to the Earthfixed frame F,. The CG has position vector r, (see Fig. 4.1), with coordinates (x,, y,, z,).
The orientation of the airplane is given by a series of three consecutive rotations, the Euler angles, whose order is important. The airplane is imagined first to be ori ented so that its axes are parallel to those of F,, (see Fig. 4.2). It is then in the posi tion Cx,y,z,. The following rotations are then applied.
1. A rotation W about oz,, carrying the axes to Cxg2z2 (bringing Cx to its final azimuth).
2. A rotation O about oy,, carrying the axes to Cx,y,z, (bringing Cx to its final elevation).
3. A rotation Q, about ox,, carrying the axes to their final position Cxyz (giving the final angle of bank to the wings).
In order to avoid ambiguities which can otherwise result in the set of angles (+, 8, 4) the ranges are limited to
Earthfixed axes. FE
Vertical
$ .,i Figure 4.2 Airplane orientation.
4.4 Orientation and Position of the Airplane 99
The Euler angles are then unique for most orientations of the vehicle. It should be noted that in a continuous steady rotation, such as rolling, the time variation of 4 for example is a discontinuous sawtooth function, and that another exception occurs in a vertical climb or dive, when 6 = + d 2 . For then ($, 8, 4) = (I) + a, ? d 2 , a) gives the same final orientation regardless of the value of a . The above difficulties can be avoided by using direction cosines3 or quaternions4 to define the orientation of the airplane instead of Euler angles. We use the Euler angles because they give a more physical picture of the airplane attitude than the other alternatives. For a more complete discussion of methods of describing vehicle orientation the reader is re ferred to Hughes (1986).
THE FLIGHT PATH
To track the flight path relative to FE, we need the velocity components in the direc tions of the axes of F,. These we get by transforming the velocity vector VE, into VE, as shown in Appendix A.4.5
Here LEE is the matrix of direction cosines that corresponds to the reverse of the se quence of rotations given above, which are for a transformation from FE to F,. Thus
where L,, L,, L, are respectively L,, L,, L, of Appendix A.4. Using the rotation matrices given in ~ ~ p e n d i x A.4, and carrying out the multiplication, we get the final result (4.4,3).
cosOcos+ sin4sinOcosrC,cos+sin* c o s ~ s i n 0 c o s ~ + s i n ~ s i n c C , cos 8 sin rC, sin 4 sin 6 sin * + cos 4 cos t,!~ cos 4 sin 0 sin i,b  sin cf~ cos cC,
sin o sin 4 cos o cos 4 cos 0 I (4.493)
The differential equations for the coordinates of the flight path are then r 7
The position of the vehicle CG is obtained by integrating the preceding equation.
S e c . 5.2 of Etkin (1972).
4Appendix E of ANSIIAIAA (1992).
'Recall that the superscript signifies velocity relative to F,, and that the subscript identifies the ref erence frame in which the components are given.
100 Chapter 4. General Equations of Unsteady Motion
ORIENTATION OF THE AIRPLANE
We now need a set of differential equations from which the Euler angles can be cal culated. These are obtained as follows: Let (i, j, k) be unit vectors, with subscripts l , 2, 3 denoting directions (x,, y , , z , ) , and so on of Fig. 4.2.
Let the airplane experience, in time At, an infinitesimal rotation from the position defined by T , 0 , @ to that corresponding to ( T + AT), (0 + A@), (@ + A@). The vector representing this rotation is approximately
and the angular velocity is exactly
An OJ = lim  = i,& + j28 + k,lk.
AIo At
By using (A.1,4), (A.4,3a) and (A.4,10) the unit vectors are found to be in frame FB
sin 8
sin 4 cos 8 cos 4 Recalling that w, = [p q rlT, we get
where
R = L O COS, s i n c p c o s ~ 1 (b) 0 sin c$ cos 4 cos 8
Inverting (4.4,6) we get the Euler angle rates as
where (4.4,7)
1 sin 4 tan 8 cos +tan 0 = [ cos 4 sin 4
0 sin 4 sec 8 cos 4 sec 8
4.5 Euler's Equations of Motion
We come now to the reason why body axes are so important. Equation (4.2,15) shows that we need the time derivative of the angular momentum, h, which contains the mo ments and products of inertia with respect to whatever axes are chosen. If those axes are fixed relative to inertial space, then the inertias will be variables in the equations. This is most undesirable and can be avoided by writing the equations in the frame FB, in which all the inertias are constant.
4.5 Euler's Equations of Motion 101
We begin with the force equation of (4.2,15):
f, = ~ v E , (4.5,1)
Both vectors in (4.5,l) are now expressed in F, components; thus:
The derivative of the transformation matrix is obtained from (A.4) as
LEB = L E B ~ B ( 4 5 3 )
With (4.5,3), (4.5,2) becomes
L,f, = rn(L,,h,V: + L ~ , v ~ )
Now premultiply by L,, to get
f , = m(V5 + h,V;) (4.54)
A similar procedure applied to the moment equation of (4.2,15) leads to
G, = h, + h,h, (4.595)
The force vector f is the sum of the aerodynamic force A and the gravitational force mg, that is,
f = m g + A (4.5,6)
where
A, = [X Y ZIT
and
mg, = mL,,g, = mL,,[O 0 glT (4.557)
We denote V ; = [uE uE wEIT and use (4.3,2b) for h,. Equations (4.5,4 and 5) are then expanded using (4.3,4) to yield the desired equations. In doing so we note that G, = [L M NIT and that, because the airplane is assumed to be rigid, I, = 0.
X  mg sin 0 = m(uE + qwE  w E ) (a)
Y + mg cos 8 sin 4 = m(vE + ru"  pwE) (b) (4.5,8)
Z + mg cos 8 cos 4 = m(wE + pvE  quE) (c>
L = I  1 ~ ( ~ ~  )  1 ( + p q )  I ~ ( ~  r p )  ( I  I ~ ) q r (a)
M = I  I  p2)  I + qr)  I  pq)  ( I  r p (b) (4.5,9)
N = 1  1 "  q 2  9 + P  P  q  ( 1  P ( c )
CHOICE OF BODY AXES
The equations derived in the preceding sections are valid for any orthogonal axes fixed in the airplane, with origin at the CG, and known as body axes. Since most air craft are very nearly symmetrical, it is usual to assume exact symmetry, and to let Cxz be the plane of symmetry. Then Cx points "forward," Cz "downward," and Cy to
102 Chapter 4. General Equations of Unsteady Motion
Principal axes Stability axes
€ > O
Figure 4.3 Illustrating two choices of body axes.
the right. In this case, the two products of inertia, I,, and I,,, are zero, and (4.5,9) are consequently simplified.
The directions of C x and C z in the plane of symmetry are conventionally fixed in one of three ways (see Fig. 4.3).
Principal Axes
These are chosen to coincide with the principal axes of the vehicle, so that the re maining product of inertia I, vanishes; (4.3,4 and 5) then yield
Stability Axes
These are chosen so that C x is aligned with V in a reference condition of steady symmetric flight. In this case, the reference values of v and w are zero, and the axes are termed stability axes. These axes are commonly used, owing to the simplifica tions that result in the equations of motion, and in the expressions for the aerody namic forces.
With this choice, it should be noted that for different initial flight conditions the axes are differently oriented in the airplane, and hence the values of I,, I,, and I, will vary from problem to problem. The "stability axes," just as the principal axes, are body axes that remain fixed to the airplane during the motion considered in any one problem.
The following formulas are convenient for computing I,, I,, I, when the values Ixp and Izp are known for principal axes (see Exercise 4.4).
1 1, =  2 (Izp  ZIP) sin 2~
where E = angle between x, (principal axis) and x, (stability axis), positive as shown (see Fig. 4.3).
4.7 The Equations Collected 103
Body Axes
When the axes are neither principal axes nor stability axes, they are usually called simply body axes. In this case the x axis is usually fixed to a longitudinal refer ence line in the airplane. These axes may be the most convenient ones to use if the aerodynamic data have been measured by a windtunnel balance that resolves the forces and moments into bodyfixed axes rather than tunnelfixed axes.
4.6 Effect of Spinning Rotors on the Euler Equations
In evaluating the angular momentum h (see Sec. 4.3) it was tacitly assumed that the airplane is a single rigid body. This is implied in the equation for the velocity of an element (4.3,2). Let us now imagine that some portions of the airplane mass are spin ning relative to the body axes; for example, rotors of jet engines, or propellers. Each such rotor has an angular momentum relative to the body axes. This can be computed from (4.3,4) by interpreting the moments and products of inertia therein as those of the rotor with respect to axes parallel to Cxyz and origin at the rotor mass center. The angular velocities in (4.3,4) are interpreted as those of the rotor relative to the air plane body axes. Let the resultant relative angular momentum of all rotors be h', with components (hi, h:, hi) in FB, which are assumed to be constant. It can be shown that the total angular momentum of an airplane with spinning rotors is obtained simply by adding h' to the h previously obtained in Sec. 4.3 (see Exercise 4.5). The equation that corresponds to (4.3,4) is then6
Because of the additional terms in the angular momentum, certain extra terms appear in the righthand side of the moment equations, (4.5,9). Those additional terms, known as the gyroscopic couples, are
In the L equation: qh:  rh: In the M equation: rhi  phi In the N equation: phi  qh:
As an example, suppose the rotor axes are parallel to Cx, with angular momentum h' = i l a . Then the gyroscopic terms in the three equations are, respectively, 0, I a r , and 1flq.
4.7 The Equations Collected
The kinematical and dynamical equations derived in the foregoing are collected be low for convenience. The assumption that Cxz is a plane of symmetry is used, so that I,, = I,, = 0, and (4.6,2) are added to (4.5,9) to give (4.7,2).
'Note that the inertias of the rotors are also included in I,.
104 Chapter 4. General Equations of Unsteady Motion
X  mg sin 6 = m(uE + qwE  ruE)
Y + mg cos 6 sin 4 = m(vE + ruE  p ~ E )
Z + mg cos 6 cos 4 = m(wE + P ~ E  quE)
L = I,@  I,,? + qr(I,  I,)  I,pq + qhi  rhi
M = I,q + rp(Ix  I,) + I , ( ~ ~  12) + rh:  phi,
N = I,?  Imp + pq(I,  Ix) + IzJqr + phl  qh:
p = d  $ s i n 6
q = 8cos 4 + $cos 6sin 4 r = CCOS 6cos 4  6sin 4 4 = p + (q sin 4 + r cos 4) tan 8
6 = q cos 4  r sin 4 $ = (q sin 4 + r cos 4 ) sec 6
kE = uE cos 6 cos I/J + vE(sin C$ sin 6 cos I/J  cos 4 sin I/J) + wE(cos 4 sin 6 cos cC, + sin 4 sin I))
YE = uE cos 6 sin cC, + vE(sin 4 sin 8 sin I/J + cos 4 cos cC,)
+ wE(cos 4 sin 6 sin cC,  sin 4 cos cC,)
iE= uEsin 6 + vEsin+cos 8 + wEcos+cos 6
uE = U + W,
v E = v + W,
wE = W + W,
4.8 Discussion of the Equations
The above equations are quite general and contain few assumptions. These are:
1. The airplane is a rigid body, which may have attached to it any number of rigid spinning rotors.
2. Cxz is a plane of mirror symmetry.
3. The axes of any spinning rotors are fixed in direction relative to the body axes, and the rotors have constant angular speed relative to the body axes.
The equations of Sec. 4.7 are many and complex. They consist of 15 coupled nonlinear ordinary differential equations in the independent variable t, and three alge braic equations. Before we can identify the true dependent variables, however, we must first consider the aerodynamic forces (X,Y,Z) and moments (L,M,N). It is clear that these must depend in some manner on the relative motion of the airplane with re spect to the air, given by the linear and angular velocities V and u, on the control variables that fix the angles of any moveable surfaces and on the settings of any propulsion controls that determine the thrust vector. Thus it is universally assumed in flight dynamics that the six forces and moments are functions of the six linear and angular velocities (u,v,w,p,q,r) and of a control vectol: The latter clearly depends to
4.8 Discussion of the Equations 105
some extent on the particular airplane, but we can, with adequate generality, write it as
of which the first three are the familiar aileron, elevator, and rudder angles, and the last is the throttle control. Other components can be added to the control vector as needed to meet special requirementsfor example, direct lift control. The control variables, from the standpoint of the mathematical system, are arbitrary functions of time. How they are determined is the subject of later sections. The wind vector whose components appear in (4.73) would ordinarily be a known function of position r, with its components given in frame F,. Its components in F, are W , = LBEW,.
The true implicit dependent variables of the system are thus seen to be 12 in number:
CG position: x,, y,, zE
Attitude: $, 6, 4 Velocity: uE, vE, wE
Angular velocity: p, q, r
Of the 15 differential equations we note that 3 of (4.7,3) are not independent, so the number of independent equations is actually 12, the same as the number of de pendent variables. The mathematical system is therefore complete.
A useful view of the equations is given in the block diagram of Fig. 4.4, which is specialized to the case of zero wind. Each block represents one set of equations, with inputs and outputs (the dependent variables). All the inputs needed for the lefthand side are generated as outputs on the right, except for the control inputs, which remain to be specified. The nature of the mathematical problem that ensues is very much
R4tr Control forces
u, v, w
Pa '7. r Control moments
49 0. * Figure 4.4 Block diagram of equations for vehicle with plane of symmetry. Body axes. FlatEarth approximation. No wind.
106 Chapter 4. General Equations of Unsteady Motion
governed by the specifics of the control inputs. In the following paragraphs, we dis cuss the various cases that commonly occur in engineering practise and in research.
STABILITY PROBLEMSCONTROLS FIXED
In these problems the airplane is considered to be disturbed from an initially steady flight condition, with the controls locked in position. Thus, c is zero or a known con stant. The equations are nonlinear and, consequently, extremely difficult to deal with analytically. In many problems of practical importance, it is satisfactory to linearize the equations by dealing only with small perturbations from the reference condition. In that case we obtain a set of homogeneous linear differential equations with con stant coefficients, a type that is readily solved. Problems of this class are treated in Chap. 6.
STABILITY PROBLEMSCONTROLS FREE
The freecontrol case is of interest primarily only for manually controlled airplanes. In that case, one or more of the primary control systems is presumed to be freed as in "hands off" by the pilot. The variation of the control angles with time, which is of course needed for the aerodynamic force and moment inputs, is then the result of an interaction between the dynamics and aerodynamics of the vehicle and those of the control system itself, which is usually simplified as a system with one degree of free dom relative to F,. For a derivation of these equations see Etkin (1972, Sec. 11.3). Each such free control adds one dependent variable and one equation to the mathe matical system.
STABILITY PROBLEMSAUTOMATIC CONTROLS
When the airplane is under the control of an AFCS the controls are neither fixed nor free. The control vector c, which fixes the control inputs of Fig. 4.4, are then deter mined by the feedback loop that activates the control systems in response to the val ues of the 12 dependent variables and inputs from other sources such as navigation, guidance, or fire control systems, and so on. Problems of this class are studied in Chap. 8.
RESPONSE TO CONTROLS
The effectiveness of an airplane's controls is conventionally studied by specifying the variation of (a,, S,, S,, 6,) with time arbitrarily, e.g., a stepfunction input of aileron angle. The airplane equations of motion then become inhomogeneous equations for (u, v, w), ( p , q, r), (0, 4, t,!r) with the control angles as forcing functions. Problems of this type are treated in Chap. 7 .
RESPONSE TO ATMOSPHERIC TURBULENCE
The motion of an airplane, and the forces that act on it, as a consequence of the tur bulent motion of the atmosphere, are very important for both design and operation. The associated mathematical problems are treated with the same general equations as
4.9 The SmallDisturbance Theory 107
given above. (uE, vE, wE) then have to be sums of (u, v, w) and the velocity of the at mosphere at the CG, and additional complications arise from the fact that the relative wind varies, in general, from point to point on the airplane. This case is treated in depth in Chap. 13 of Etlun (1972) and in Etkin (1981).
INVERSE PROBLEMS
A class of problems that has not received much attention in the past, but that is never theless both interesting and useful, is that in which some of the 12 variables usually regarded as dependent are prescribed in advance as functions of time. An equal num ber of equations must then be dropped in order to maintain a complete system. This is the kind of problem that occurs when we ask questions of the type "Given the air plane motion, what pilot action is required to produce it?'Such questions may be rel evant to problems of control design and maneuvering loads.
The mathematical problem that results is generally simpler than those of stability and control. The equations to be solved are sometimes algebraic, sometimes differen tial. A decided advantage is the ability of this approach to cope with the nonlinear equations of large disturbances.
Another category is the mathematical problem that arises in flight testing when time records are available of some control variables and some of the 12 dependent variables. The question then is "What must the airplane parameters be to produce the measured response from the measured input?' (See Etkin, 1959, Chap. 11; AGARD 1991; Maine and Iliff, 1986.) This is an example of the important "plant identifica tion" problem of system theory.
4.9 The SmallDisturbance Theory
As remarked in Sec. 4.8, the equations of motion are frequently linearized for use in stability and control analysis. It is assumed that the motion of the airplane consists of small deviations from a reference condition of steady flight. The use of the smalldis turbance theory has been found in practice to give good results. It predicts with satis factory precision the stability of unaccelerated flight, and it can be used, with suffi cient accuracy for engineering purposes, for response calculations where the disturbances are not infinitesimal. There are, of course, limitations to the theory. It is not suitable for solutions of problems in which large disturbance angles occur, for ex ample @ = d 2 .
The reasons for the success of the method are twofold: (1) In many cases, the major aerodynamic effects are nearly linear functions of the disturbances, and (2) dis turbed flight of considerable violence can occur with quite small values of the linear and angularvelocity disturbances.
NOTATION FOR SMALL DISTURBANCES
The reference values of all the variables are denoted by a subscript zero, and the small perturbations by prefix A. When the reference value is zero, the A is omitted. All the disturbance quantities and their derivatives are assumed to be small, so that their squares and products are negligible compared to firstorder quantities.
108 Chapter 4. General Equations of Unsteady Motion
The reference flight condition is assumed for convenience to be symmetric and with no angular velocity. Thus v, = p, = q, = r, = 4, = 0. Furthermore, stability axes are selected as standard in this book, and thus w, = 0 for all the problems con sidered. u, is then equal to the reference flight speed, and 8, to the reference angle of climb (not assumed to be small). In dealing with the trigonometric functions in the equations the following relations are used:
sin (8, + A8) = sin $ cos A8 + cos 0, sin A0
= sin 6, + A0 cos 8,
cos (8, + A8) = cos 8, cos A8  sin 8, sin A8
= cos 6,  A8 sin 8,
FURTHER ASSUMPTIONS
The smalldisturbance equations will be slightly restricted by the adoption of two more assumptions, which correspond to current practice. These are
1. The effects of spinning rotors are negligible. This is the case when the air plane is in gliding flight with power off, when the symmetrical engines have opposite rotation, or when the rotor angular momentum is small.
2. The wind velocity is zero, so that VE = V
LINEARIZATION
When the smalldisturbance notation is introduced into the equations of Sec. 4.7, the additional assumptions noted above are incorporated, and only the firstorder terms in disturbance quantities are kept, then the following linear equations are obtained.
X, + AX  mg(sin 8, + A8 cos $) = mAu
Yo + AY + mg4 cos $ = m(v + u,r)
Zo + AZ + mg(cos $  A8 sin $0) = m(w  uoq)
Lo + AL = I,p  I,i
M, + AM = I,q
No + Ah' = 1,p + I,?
o = ~ + = p + r t a n O 0 , p = +  & s i n $
$ = r S ~ C 6,
RE = (u, + Au) cos 8,  uoA8 sin 80 + w sin 8,
Y,= U,+COS eo+ v iE = (u0 + Au) sin 8  uoA8 cos 8, + w cos 8,
REFERENCE STEADY STATE
If all the disturbance quantities are set equal to zero in the foregoing equations, then they apply to the reference flight condition. When this is done, we get the following
4.9 The SmallDisturbance Theory 109
relations, which may be used to eliminate all the reference forces and moments from the equations:
X,,  mg sin 0, = 0
Yo = 0
Zo + mg cos 8, = 0 (4.9,6) L = M = N = O
0 0 0
kE, = u, cos O,,, jE,, = 0, iE,, = uo sin 8,
We further postulate that in the reference steady state the aileron and rudder angles are zero. When (4.9,6) are substituted into (4.9,2 to 4 . 9 3 , (4.9,3) are solved for p and i, and the equations are rearranged, the result is (4.9,7 to 4.9,10).
AY zi =  + g+ cos 0,  uor
m
i = (IJz  1;)'(1AL + IXw
4 = p + r tan 0,; p = 4  4 sin 0, * = r sec 0,
AiE = Au cos 0,  uoAO sin 0, + w sin 6,
AjE = uO$ cos 80 + v A& = Au sin 8,  u0A8cos 0, + w cos 0,
THE LINEAR AIR REACTIONS
At the heart of the subject of atmospheric flight mechanics lies the problem of deter mining and describing the aerodynamic forces and moments that act on a given body in arbitrary motion. It is primarily this aerodynamic ingredient that distinguishes it from other branches of mechanics. Aerodynamic forces and moments are strictly speaking functionals of the state variables. Consider for example the timedependent lift L(t) on a wing with variable angle of attack a(t). Because the wing leaves behind it a vortex wake that in general generates an induced velocity field at the wing, and because hysteresis is present in flow separation processes, the aerodynamic field that fixes the lift at any given moment is actually dependent not only on the instantaneous value of a but strictly speaking on its entire past history. This functional relation is expressed by
110 Chapter 4. General Equations of Unsteady Motion
When a(r ) can be expressed as a convergent Taylor series around t, i.e.
then the infinite series a(t), iu(t), Li(t)    can replace a ( ~ ) in (4.9,l I), i.e.
L(t) = L(a, iu, Li  ) (4.9,13)
where a , iu  0  are values at time t. Thus the lift at time t is in this case determined by a and all its derivatives at time t. A further series expansion of the righthand side of (4.9,13) around t = 0 yields
in which all the products and powers of Aa, Aiu ... appear, and where
etc.
The classical assumption of linear aerodynamic theory, due to Bryan (191 1 ) is to ac cept the linear reduction of (4.9,14) as a representation of the aerodynamic force, even when Aa(t) is not an analytic function as implied by (4.9,12), i.e.
Derivatives such as La in (4.9,16) are known as the stability derivatives, or more gen erally as aerodynamic derivatives. For most forces and state variables, only the first term of (4.9,16) is kept, but in some cases, terms up to the second derivative must be retained for sufficient accuracy. This assumption has been found to work extremely well over a wide range of practical applications. Occasionally the addition of nonlin ear terms such as %,,(Aa)' = L,,(A~)' can extend the useful range considerably. Another way of including nonlinear effects is to treat the derivatives as functions of the variables, for example, L, = La(&).
A major fraction of the total effort in aerodynamic research in the past has been devoted to the determination, by theoretical and experimental means, of the aerody namic derivatives needed for application to flight mechanics. A great mass of infor mation about these parameters has now been accumulated and Chap. 5 is devoted to this topic.
For a truly symmetric configuration, it is evident that the side force Y, the rolling moment L, and the yawing moment N will all be exactly zero in any condition of symmetric flight, that is, when the plane of symmetry remains in a fixed vertical plane. In that case, P, p, r, 4, +, and y, are all identically zero. Thus the derivatives of the asymmetric or lateral forces and moments, Y, L, N with respect to the symmet ric or longitudinal motion variables u, w, q are zero. In writing out the complete lin ear expression for the aerodynamic forces and moments, we use this fact, and in ad dition make the further approximations:
1 . We may neglect as well all the derivatives of the symmetric forces and mo ments with respect to the asymmetric motion variables.
2. We may neglect all derivatives with respect to rates of change of motion vari ables except for 2, and M,.
4.9 The SmallDisturbance Theory 111
3. The derivative Xq is also negligibly small.
4. The density of the atmosphere is assumed not to vary with altitude (see Sec. 6.5).
It should be emphasized that none of these assumptions is basically necessary for the solution of airplane dynamics problems. They are made as a matter of experience and convenience. When it appears necessary to do so, any of the terms dropped can be restored into the equations. With these assumptions, however, the linear forces and moments are:
AX = X,Au + Xww + AX, (a) AY = Y,v + Ypp + Yrr + AY, (b) AZ = ZuAu + Zww + Z,w + Zqq + AZ,
AL = L,v + Lpp + L,r + AL, (4.9,17)
(dl AM = MuAu + M,w + M,w + Mqq + AM, (el AN = N,v + N,p + Nrr + AN, ( f )
In the above equations, the terms on the right with subscript c are control forces and moments that result from the control vector c. Explicit forms for the controls will be introduced as they are needed in the following.
Aerodynamic Transfer Functions
The preceding equations are subject to the theoretical objection (not of great practical importance) that the Bryan formulation for the aerodynamics is not quite sound even within the restriction of linearity. This is readily illustrated by considering the lift on a wing following a step change in angle of attack. Let Aa be given by Aa(t) = a,l(t) where a, is a constant. For t > 0, the Bryan formula (4.9,16) gives
whereas in fact the lift undergoes a transient approach to the asymptote La%, the de tails of which depend on the wing shape and the Mach number. Equation (4.9,16) fails in this case because Aa is not an analytic function, having a discontinuity at t = 0. Now the transient process is often well approximated as a linear one, and as such is subject to exact representation by linear mathematics in the form of an indi cia1 function (Tobak, 1954), or an aerodynamic transfer function (Etkin, 1956). The implementation of these alternative representations of aerodynamic force is described in Etkin (1972, Sec. 5.1 1).
THE LINEAR EQUATIONS OF MOTION
When (4.9,17) are substituted into (4.9,7 to 4.9,10) two of the equations [(4.9,7c) and (4.9,86)], contain w terms on the righthand side. In order to retain the desired form with first derivatives of the dependent variables on the left, we have to solve these two equations simultaneously for w and q. The result is presented in matrix form in (4.9,18 and 4.9,19). Here the equations are divided into two groups, termed longitu dinal and lateral, for reasons that are explained as follows.
114 Chapter 4. General Equations of Unsteady Motion
As a consequence of the simplifying assumptions made in their derivation, the pre ceding equations have the exceedingly useful property of splitting into two indepen dent groups. Suppose that 4, v , p, r, AY,, AL, and m, are identically zero. Then (4.9,19) are all identically satisfied. The remaining equations (4.9,18) form a com plete set for the six homogeneous variables Au, w, q, A8, Ax,, Az,. Thus we may conclude that modes of motion are possible in which only these variables differ from zero. Such motions are called longitudinal or symmetric, and the corresponding equa tions and variables are likewise named. Conversely, if the longitudinal variables are set equal to zero, the remaining six equations (4.9,19) form a complete set for the de termination of the variables 4, +, v, p, r, y,. These are known as the lateral variables, the corresponding equations and motions being likewise named.
It is worthwhile recording here the specific assumptions upon which this separa tion depends. A study of the various steps that have led to the final equations reveals these factsthe existence of the pure longitudinal motions depends on only two as sumptions:
1. The existence of a plane of symmetry. 2. The absence of rotor gyroscopic effects.
The existence of the pure lateral motions, however, depends on more restrictive ap proximations; namely
1. The linearization of the equations. 2. The absence of rotor gyroscopic effects.
3. The neglect of all aerodynamic crosscoupling (approximation 1 p. 110).
If the equations were not linearized, then there would be inertial crosscoupling between the longitudinal and lateral modes, as evidenced by terms such as mpvE in (4.7,lc) and rp(I,  I,) in (4.7,2b). That is, motion in the lateral modes would induce longitudinal motion.
Equations (4.9,18 and 4.9,19) are both in the desired firstorder form, commonly referred to as state vector form, conventionally written in vectorlmatrix notation as
Here x is the state vector, c is the control vector, and A and B are system matrices. The state vectors for the longitudinal and lateral systems are, respectively:
and the matrices A for the two cases can be inferred from the full equations. The de pendent variables x,, y,, z, and JI are not included in the state vectors because they do not appear on the righthand side of the equations. The matrix B will be discussed later when we come to the analysis of controlled motions.
4.10 Nondimensional System 115
4.1 0 Nondimensional System
The reader will already be familiar with the great advantage of using nondimensional coefficients for aerodynamic forces and moments such as lift, drag, and pitching mo ment. In this way the major effects of speed, size, and air density are automatically accounted for. Similarly we need nondimensional coefficients for the many deriva tivesXu and so onthat occur in (4.9,18 and 4.9,19). Unfortunately, there is no universally accepted standard for these coefficients, although attempts have been made to devise one (e.g., ANSUAIAA, 1992). The student, and indeed the practising engineer, should be sure to note carefully the exact notation and definitions employed in any reference material or data sources being used. The notation and definitions used in this book are essentially the NASA system, which is widely used.
Before presenting this system, we digress briefly to a dimensional analysis of the general flight dynamics problem. This helps to provide insight into what the true un derlying variables are, and provides a basis for what follows. Imagine a class of geo metrically similar airplanes of various sizes and masses in steady unaccelerated flight at various heights and speeds. Suppose that one of these airplanes is subjected to a disturbance. After the disturbance, some typical nondimensional variable .rr varies with time. For example, .rr may be the angle of yaw, the load factor, or the helix angle in roll. Thus, for this one airplane, under one particular set of conditions we shall have
Let it be assumed that this equation can be generalized to cover the whole class of airplanes, under all flight conditions. That is, we shall assume that .rr is a function not o f t alone, but also of
where m is the airplane mass and 1 is a characteristic length. Instead of (4.10,1), then, we write
Buckingham's .rr theorem (Langhaar, 1951) tells us that, since there are nine quanti ties in (4.10,2) containing three fundamental dimensions, L, M, and T, then there are 9  3 = 6 independent dimensionless combinations of the nine quantities. These six socalled .rr functions are to be regarded as the meaningful physical variables of the equation, instead of the original nine. Two systems of the same class are dynamically similar when all the .rr functions of one are numerically equal to those of the other. By inspection, we can easily form the following six independent nondimensional combinations:
m u,t uz T, M, RN, , , 
1 lg
Following the .rr theorem, we write as the symbolic solution to our problem
m u,t ug M , RN, , , 
pl' 1 lg
116 Chapter 4. General Equations of Unsteady Motion
The effects of the six variables m, p, 1, g, u,, and t are thus seen to be compressed into the three combinations: mlp13, u$lg, and u,tll. We replace l3 by Sl, where S is a char acteristic area, without changing its dimensions, and denote the resulting nondimen sional quantity mlpS1 by p.. The quantity llu, has the dimensions of time and is de noted t*. The quantity ugllg is the Froude number (FN). Equation 4.10,3 then becomes
.rr = f (M, RN, FN, p., t/t*) (4.10,4)
The significance of (4.10,4) is that it shows .rr to be a function of only five vari ables, instead of the original eight. The result is of sufficient importance that it is cus tomary to elaborate on it still further. Since p. is the ratio of the airplane mass to the mass of a volume Sl of air, it is called the relative mass parameter or relative density parametel: It is smallest at sea level and increases with altitude.
The main symbols for which nondimensional forms are wanted are listed in Table 4.1. The nondimensional item in column 3 is obtained by dividing the corre sponding dimensional item of column 1 by the divisor in column 2. In the smalldis turbance case, ir and 6 are aerodynamic angles, for then
V = [(u, + Au)' + u2 + w ~ ) ] " ~
From (1.6,4)
ax = tanP'(w/u) = tan'[wl(u, + Au)]
To first order in v, w, and Au these are
Table 4.1 The Nondimensional System
(1 ) (2 ) (3)
Divisor Divisor Small
Dimensional General Disturbance Nondimensional Quantity Case Case Quantity
Note: (1) p and Aa are used interchangeably with O and I?, respectively, in the small disturbance case.
4.10 Nondimensional System 117
Table 4.2 Longitudinal Nondimensional Derivatives
and similarly
NONDIMENSIONAL STABILITY DERIVATIVES
The nondimensional stability derivatives are the partial derivatives of the force and moment coefficients in lines 1, 3, and 4 of Table 4.1 with respect to the nondimen sional motion variables in lines 5 , 6, and 7. The notation for these is displayed in Ta bles 4.2 and 4.3. Each entry in the tables represents the derivative of the column heading with respect to the row variable.
Since ax differs from a only by a constant (the angle between the zerolift line and the x axis), then A a x = A a , alas;,  d / a a , and no distinction need be made be tween these two derivatives.
NONDIMENSIONAL EQUATIONS
It is possible with the definitions given in Tables 4 .14 .3 to make the equations of motion entirely nondimensional, and such equations have been widely used in the past, especially for analytical work (see Etkin, 1972 and 1982). The prevailing cur rent practise in design and research, however, is to use the dimensional equations and program them for calculation on a digital computer. We are therefore not including the nondimensional equations in this book. There is no real loss in so doing however, since any analytical results that are obtained with the dimensional equations can sub sequently be expressed, for maximum generality, in nondimensional form. Examples of this are contained in Chaps. 6 and 7.
Table 4.3 Lateral Nondimensional Derivatives
c,, Cl cn
118 Chapter 4. General Equations of Unsteady Motion
Table 4.4 Longitudinal Dimensional Derivatives
u pu$C,, sin 0, + ipu,JCXU  pu$C,, cos oo + ipu,JCZu +puO~SCmu
W ~PUOSC~, afu,JCz, ipu0~SCma 9 ipuocscxq tpu0cscz, fpu0c2SCm, w ipzsc,, f ~ S C , , fpz2SCm,
4.1 1 Dimensional Stability Derivatives
We now need expressions for the derivatives that appear in (4.9,18) and (4.9,19) in terms of the nondimensional derivatives. A few examples of these are derived as fol lows to illustrate the procedure, and the whole set needed for (4.9,17) is displayed in Tables 4.4 and 4.5. Derivatives with respect to D or B are usually negligible and are not included.
THE Z DERIVATIVES
From Table 4.1, Z = c,ipv2~ where v2 = u2 + v2 + w2 and u = u, + Au. Hence
where the subscript zero indicates the reference flight condition. But 2V(aVIau) = 224, and hence (avlau), = 1. Also
a cz 1 ac, 1 = ; (=lo = & czu
From (4.9,6)
Z, = mg cos 8,
hence
c,, =  c,, cos eo so that
Table 4.5 Lateral Dimensional Derivatives
Y L N
u apuoSCyp h p ~ , b S C , ~ ipu&SC,, P f puobSCyP f p ~ , b ~ s c , ~ fpu,b2scnP r tpuobSCyr ipuob2SC,r +puob2SC,r
4.1 1 Dimensional Stability Derivatives 11 9
Also, since (avla~)~, = 0, then
But w = u,cu,, so that
In a similar way,
But
Hence
Also,
But
Hence
THE X DERIVATIVES
These are found in a manner similar to the Z derivatives. In this instance from (4.9,6)
Cxc, = Cwo sin 8"
THE M DERIVATIVES
These are also found in a manner similar to the Z derivatives. In this case we start with M = C,Bpv2Sc and note from (4.9,6) that C,,, = 0.
120 Chapter 4. General Equations of Unsteady Motion
THE L DERIVATIVES
b From Table 4.1, L = ClpV2S . Hence
2
Also
Similarly
a~ b ac, b2 ac, L p = (=&), = p u 3 (&JO = ~ ~ 0 s Q (%) 0
1 =  puob2SClr
4
THE N DERIVATIVES
These are found in a manner similar to the L derivatives.
THE Y DERIVATIVES
These are also found in a manner similar to the L derivatives. In this case we start with Y = CY$pv2S.
4.12 Elastic Degrees of Freedom
In the preceding sections we have presented the "main" equations of motion, that is, those associated with the six rigidbody degrees of freedom. Now it is known that the stability and control characteristics of flight vehicles may be profoundly influenced by the elastic distortions of the structure under aerodynamic load (AGARD, 1970; Milne, 1964; McLaughlin, 1956; Rodden, 1956). Additionally, there are phenomena not primarily related to stability and control, but rather to structural integrity, in which elastic deformation is a primary elementi.e., structural divergence and flut ter. In order to understand and analyze all these effects, one needs the equations that govern the elastic deformations, and as well the changes that such deformation intro duces into the six main dynamical equations.
A full treatment of this branch of flight mechanicsaeroelasticity and structural vibrationis beyond the scope of this text, and the reader is referred to (Bispling hoff, 1962 and Dowell, 1994) for comprehensive treatises on it. Here we content our selves with presenting the framework of the analysis, but omit most of the structural
4.12 Elastic Degrees of Freedom 121
and aerodynamic details. Enough material is given, however, to show how the static and dynamic deformations are integrated into the preceding mathematical model of the "gross" vehicle motion.
The deformation analysis is almost invariably treated by a linear theory, even when the rigidbody motion is not. We shall therefore assume at the outset that the distortional motions are "small" and that all the associated aerodynamic forces are linear functions.
THE METHOD OF QUASISTATIC DEFLECTIONS
Many of the important effects of distortion can be accounted for simply by altering the aerodynamic derivatives. The assumption is made that the changes in aerody namic loading take place so slowly that the structure is at all times in static equilib rium. (This is equivalent to assuming that the natural frequencies of vibration of the structure are much higher than the frequencies of the rigidbody motions.) Thus a change in load produces a proportional change in the shape of the vehicle, which in turn influences the load. Examples of this kind of analysis are given in Sec. 3.5 (ef fect of fuselage bending on the location of the neutral point), and Secs. 5.3 and 5.10.
THE METHOD OF NORMAL MODES
When the separation in frequency between the elastic degrees of freedom and the rigidbody motions is not large, then significant inertial coupling can occur between the two. In that case a dynamic analysis is required, which takes account of the time dependence of the elastic motions.
The method that is described here for accomplishing this purpose is based upon the representation of the deformation of the elastic vehicle in terms of its normal modes of free vibration. Imagine that the vehicle is at rest under the action of no ex ternal forces, aerodynamic, gravity, or other, and that a frame of reference with origin at the mass center, but otherwise arbitrary, is attached to it. The position of mass ele ment 6m is then (x,, yo, 2,). Now let the structure be deformed by a selfequilibrating set of external forces and couples, so that it takes a new form, stationary with respect to the coordinate system. Upon instantaneous release of this force system, a free vi bration ensues, that is, one in which external forces play no part, and in which the po sition of 6m at time t is (x, y, z). Since there is zero net force, and zero net moment, the linear and angular momenta of the elastic motion must vanish, whatever the ini tial distorted shape. In particular this is true for each and every undamped normal mode of free vibration. Any small arbitrary elastic motion of the vehicle can, there fore, relative to the chosen axes (transients as well as steady oscillations), be repre sented by a superposition of free undamped normal modes as follows:
m
yl(t) = 1 1 g,(x,, yo, zo)€,(t)
122 Chapter 4. General Equations of Unsteady Motion
where (x', y', z ' ) are the elastic displacements, (x  x,,) etc., (f,, g,, h,) are the mode shape functions: and ~, ( t ) are the generalized coordinates giving the magnitudes of the modal displacements.
We have specified idealized undamped modes, as opposed to the true modes of a real physical structure with internal and external damping, because the latter may not be "simple" modes with fixed nodes, describable by a single set of three functions. More generally they each consist of a superposition of two "submodes" 90" out of phase. Because of this, the equations of motion for the elastic degrees of freedom of the real structure are not perfectly uncoupled from one another, but contain intercou pling damping terms that would usually be negligible in practical applications.
The use of the free undamped normal modes is seen to ensure that the linear and angular momenta of the distortional motion vanishes. Consequently the elastic mo tions have no inertial coupling with the rigidbody motions except through the mo ments and products of inertia. However, it can be shown that this coupling is second order and negligible in the smallperturbation theory. The determination of the shapes and frequencies on of the normal modes is a major task, and the methods for finding them are beyond the scope of this text. For treatises on this subject the reader should refer to (Bisplinghoff et al., 1955; Fung, 1955). As indicated in (4.12,1), there are ac tually an infinite number of normal modes of vehicle structures. In practice, of course, only a finite number N of those at the lowfrequency end of the set need be retained, and the summations in (4.12,l) are approximated by finite series of N terms. Some judgment and experience is needed to decide just how many modes are needed in any application, but a general rule that is helpful is to discard those whose frequen cies are substantially higher than the significant ones present in the spectral represen tation of inputs arising from control action or atmospheric turbulence.
MODIFICATION OF THE RIGID BODY EQUATIONS
Although the inertia terms of the previous equations, for example, (4.9,18) and (4.9,19), remain unchanged to first order by the presence of elastic motions, the elas tic and rigidbody motions are not nevertheless entirely uncoupled.
The deformations of the structure in general cause perturbations in the aerody namic forces and moments. These may be introduced into the linearized equations of motion by the addition of appropriate derivatives to the expressions for the aerody namic forces given by (4.9,17). For example, the added terms in the pitching moment associated with the nth elastic degree of freedom would be
M,"E, + Me,€, + Mzvi!, (4.12,2)
Similar expressions appear for each of the added degrees of freedom, and in each of the aerodynamic force and moment equations. An example of the elastic stability de rivatives is given in Sec. 5.10. Alternatively, the aerodynamic forces may be formu lated in the form of transfer functions.
THE ADDITIONAL EQUATIONS OF MOTION
The additional equations are most conveniently found by using Newton's laws as ex pressed by Lagrange's equations of motion (Synge and Griffith, 1942) with the E, as
'The eigenfunctions of the linear vibration problem.
4.12 Elastic Degrees of Freedom 123
generalized coordinates. The appropriate form of Lagrange's equation for this appli cation is
where T is the kinetic energy of the elastic motion relative to FB, U is the elastic strain energy, and 9, is the generalized external force. Since the coordinates are mea sured in the frame F,, which is nonNewtonian by virtue of its general motion, an ap propriate modification must be made to the external force field acting on the system when calculating the generalized force. This consists of adding to each element of mass Sm an inertial body force equal to  a' Sm where a' is that part of the total ac celeration of Sm that arises from the acceleration and rotation of FB (see Appendix A.6).
Since normal modes have been chosen as the degrees of freedom, then the indi vidual equations of motion are independent of one another insofar as elastic and iner tia forces are concerned (this is a property of the normal modes), although the equa tions will be coupled through the aerodynamic contributions to the %'s. The lack of elastic and inertia coupling permits the lefthand side of (4.12,3) to be evaluated by considering only a single elastic degree of freedom to be excited. Let its generalized coordinate be 6,. The kinetic energy is given by
where the integration is over all elements of mass of the body. From (4.12,l) this be comes (with only E, excited)
T = t i t I (f: + g: + hi) dm
The integral is the generalized inertia in the nth mode, and is denoted by
I,, = (ft + g; + hi) dm I so that
The first term of (4.12,3) is therefore I,,€,, and the second term is zero. The strainenergy term is conveniently evaluated in terms of the natural fre
quency of the nth mode by applying Rayleigh's method. This uses the fact that, when the system vibrates in an undamped normal mode, the maximum strain energy occurs when all elements are simultaneously at the extreme position, and the kinetic energy is zero. This maximum strain energy must be equal to the maximum kinetic energy that occurs when all elements pass simultaneously through their equilibrium position, where the strain energy is zero. Hence, if 6, = a sin o,t, then the maximum kinetic energy is, from (4.12,5)
124 Chapter 4. General Equations of Unsteady Motion
Since the stressstrain relation is assumed to be linear, the strain energy8 is a qua dratic function of en; that is, U = i k e . Hence
Umax = $ka2 = Tmax = $Inw;a2
It follows that k = Inw;, and that
U = 4znw;<
and hence aUlae,, = Inw;en. The left side of (4.12,3) is therefore as follows:
In€,, + Inw;en = %,, (4.12,6)
When structural damping is present, this simple form of uncoupled equation is not exact but the changes in frequency and mode shape for small damping are not large. Hence damping can be allowed for approximately by adding a damping term to (4.12,6), that is,
€, + 2[,,wnin + @;en = %,,/I, (4.12,7)
without changing wn or the modeshape functions. The value of 5 is ordinarily less than 0.1, and usually must be found by an experimental measurement on the actual structure.
EVALUATION OF Sn
The generalized force is calculated from the work done during a virtual displacement,
where W is the work done by all the external forces, including the inertia forces asso ciated with nonuniform motion of the frame of reference. The inertia force field is given by
df i = (af) d m (4.12,9)
where the components of the r.h.s. in FB are given by (A.6,8) without the terms (x, jj, f). The work done by these forces in a virtual displacement of the structure is
where the integration is over the whole body. Introducing (4.12,l) this becomes
whence ( f n df,, + gn df,, + hn dfzi) (4.12,lO)
When the inertiaforce expressions are linearized to small disturbances, and substi tuted into (4.12,10), all the remaining firstorder terms contain integrals of the fol lowing types:
8For example, in a spring of stiffness k and stretch x, the strain energy is U = +kxZ.
4.12 Elastic Degrees of Freedom 125
The first of these is zero because the origin is the mass center, and the second is zero because the angular momentum associated with the elastic mode vanishes. The net result is that 9 , = 0. This result simply verifies what was stated above; that is, there is no inertial coupling between the elastic and rigidbody degrees of freedom.
The remaining contribution to ?Pn is that of the aerodynamic forces. Let the local normalpressure perturbation at an element dS of the airplane's surface be p(x,, yo, z,), and let the local outward normal be n(n,, n,, n,). Then the work done by the aero dynamic forces in a virtual displacement is
where the integral is over the whole surface of the airplane, and ( r  r,) is the vector displacement at dS. It is given by
m
r  r, = 1 ( i f , + jg, + kh,) 6e, n= 1
hence
and
Each of the variables inside the integral is a function of (x,, yo, z,), i.e., of position on the surface, and moreover, p is in the most general case a function of all the general ized coordinates, of their derivatives, and of the controlsurface angles. The result is that 9, is a linear function of all these variables, which may be expressed in terms of a set of generalized aerodynamic derivatives (or alternatively aerodynamic transfer functions), namely,
In application, only the important derivatives would be retained in any given case. The values of the derivatives kept would be computed by application of (4.12,ll). An example of this computation is given in Sec. 5.10.
The effects of structural dynamics on the stability and control equations can be incor porated by adding structural degrees of freedom based on free normal modes. For an exact representation, an infinite number of such modes are required; however, in
126 Chapter 4. General Equations of Unsteady Motion
practice only a few of the lowest modes need be employed. The six rigidbody equa tions are altered only to the extent of additional aerodynamic terms of the type given in (4.12,2). One additional equation is required for each elastic degree of freedom (4.12,7). The generalized forces appearing in the added equations contain only aero dynamic contributions, which are computed from (4.12,ll) and expressed as in (4.12,12).
4.13 Exercises
4.1 Prove that the angular momentum vector h of an airplane is the same whether or not the wind vector W is zero.
4.2 Carry out the expansion of Phr to derive the result given in (4.3,6).
4.3 Carry out the multiplication of the three rotation matrices indicated in (4.4,2) to ob tain the result given in (4.4,3).
4.4 Derive (4.5,11) for the moments and product of inertia in frame F, when they are given for principal axes.
4.5 Prove that when an airplane has spinning rotors, such as jet engines, the angular mo , mentum is given by (4.6,l). Derive the additional terms in the moment equations
given in (4.6,2).
1 4.6 Two airplanes are geometrically similar and have similar mass distributions. Airplane I A has a span of 100 ft (30.48 m) and a weight of 100,000 Ib (445,000 N). B has 150ft
(47.72 m) span and weighs 225,000 lb (1,001,250 N). Both fly at speeds low enough to neglect Mach number effects, and high enough to neglect Reynolds number ef fects.
When flying at 400 knots and 20,000ft (6,096m) altitude, airplane B has a spi ral divergence (a lateral instability) that has a characteristic time of 20 seconds.
(a) At what speed and altitude will A be dynamically similar to B?
(b) What will be the characteristic time of the spiral divergence of A at that speed and altitude?
(c) What is the ratio of the C, values for the two flight conditions?
4.7 Substitute the linear expressions for A 2 and AM into the right side of (4.9,7c) and (4.9,8b) and solve the resulting equations to get the second and third components of (4.9,18).
4.8 Derive the expressions given in Tables 4.4 and 4.5 for the derivatives X,, M,, Y,, Nu.
4.9 Let [ p , q , rEIT be the angular velocity of an airplane in frame F,. Find a set of equa tions that produces these components, given the body axis components [ p q rIT.
4.10 A hovercraft in ground effect is acted on by the following aerodynamic forces, ex pressed as body frame components:
X = Y = O
Z = mg + ZzzE
L = L,+; M = MOB; N = 0
4.14 Additional Symbols Zntroduced in Chapter 4 127
The body axes are principal axes, and the enginehotor angular momentum is
Derive a set of smalldisturbance equations of motion.
4.11 Suppose that the wind vector W is not zero but instead is given by
where W is assumed to be "small" and a known function of time. Derive the addi tional terms that would have to be added to the smalldisturbance equations of mo tion (4.9,18) and (4.9,19).
4.12* An aircraft is performing a rolling pullup. At the instant of observation, the vehicle is at the bottom of a vertical circle of 2000 ft (610 m) radius moving at a constant speed of 500 fps (152 mls) with wings horizontal. (See Fig. 3.1). At the same time the roll rate is constant at p = 90" sI. Given that
Iy  I, = 300 slug ft2 (406 kg m2) and I, = 500 slug ft2 (677 kg m2)
determine the moments required at this time to perform this maneuver. Assume that the axes are principal axes, with Cx horizontal. (You may assume constant Euler an gle rates and rC, = 0.)
4.14 Additional Symbols Introduced in Chapter 4
mglipu,2 S
moments of inertia about (x, y, z ) axes
see (4.9,19)
product of inertia S yz dm
product of inertia S xz dm
see (4.9,19)
product of inertia $ xy dm resultant external force vector
generalized force in Lagrange's equation
resultant external moment vector, about the mass center
angular momentum vector of the airplane with respect to its mass center
angular momentum vector of spinning rotors with respect to ro tor mass center
scalar components of h in FB
scalar components of h' in FB
scalar components of G in FB
scalar components of a, radianslsec in FB
*Problem courtesy of Prof. F. H. Lutze, Virginia Polytechnic Institute
128 Chapter 4. General Equations of Unsteady Motion
(u, U , w) scalar components of V in F,
V airspeed vector of airplane mass center
(x, y, z ) components of resultant aerodynamic force acting on the air plane, in F,
x ~ , YE, ZE coordinates of airplane mass center relative to fixed axes (see Fig. 4.2)
En generalized coordinate of the nth elastic mode
a,, a,, 6, angles of elevator, rudder, and aileron
8, propulsion control angular velocity vector of the airplane
(94 0, 4) Euler angles, radians (see Sec. 4.4)
See also Tables 4.1 to 4.3.
See also Secs. 2.1 1 and 3.15.
C H A P T E R 5
The Stability Derivatives
5.1 General Remarks
We saw in Chap. 4 how the aerodynamic actions on the airplane can be represented approximately by means of stability derivatives (or more exactly by aerodynamic transfer functions). Indeed, all the aerodynamics involved in airplane dynamics is concentrated in this section of the subject: i.e., in the determination of these deriva tives (or transfer functions). Each of the stability derivatives contained in the equa tions of motion is discussed in the following sections. Wherever possible, formulas for them are given in terms of the more elementary parameters used in static stability and performance. Where this is not feasible, it is shown in a qualitative way how the particular force or moment is related to the relevant perturbation quantity. No data for estimation are contained in this chapter; these are all in Appendix B.
EXPRESSIONS FOR C, AND C,
For convenience, we shall want the derivatives of C, and C, expressed in terms of lift, drag, and thrust coefficients. The relevant forces are shown in Fig. 5.1. As shown, the thrust line does not necessarily lie on the x axis. However, the angle between them is generally small, and we shall assume it be zero. With this assumption, and for small a,, we get'
where C, is the coefficient of thrust, T / $ ~ V ~ S .
5.2 The cx Derivatives (C,, C,_, Cmn)
The a derivatives describe the changes that take place in the forces and moments when the angle of attack of the airplane is increased. They are normally an increase in the lift, an increase in the drag, and a negative pitching moment. The contents of Chap. 2 are relevant to these derivatives.
'Since X and Z are the aerodynamic forces acting on the airplane, there are no weight components in (5.1,l).
130 Chapter 5. The Stability Derivatives
W
Figure 5.1 Forces in symmetric flight.
THE DERIVATIVE Cxa
By definition, C,, = (aC,/aa),, where the subscript zero indicates that the derivative is evaluated when the disturbance quantities are zero. From (5.1,l)
ac, ac, acL ac,  + CL + a,   aa aa aa aa
We may assume that the thrust coefficient is sensibly independent of cu, so that aC,/aa = 0, and hence
where the subscript zero again indicates the reference flight condition, in which, with stability axes, a, = 0. When the drag is given by a parabolic polar in the form CD = C,,,,, + C ~ I T A ~ , then
THE DERIVATIVE Czu
By definition, Czu = (acjaa),. From (5.1,l) we get
Therefore
Czu =  (CLu + CD,) (5 .23
C,,, will frequently be negligible compared to C,,, and consequently Czu =  CLu.
THE DERIVATIVE Cmu
Cma is the static stability derivative, which was treated at some length in Chap. 2. It is conveniently expressed in terms of the stickfixed neutral point (2.3,25):
5.3 The u Derivatives (CxU, CzU, CmU) 131
cmcx = a(h  h,) (5.294)
For airplanes with positive pitch stiffness, h < h,, and Cma is negative.
5.3 The u Derivatives (Cxu, CZ,, CmU)
The u derivatives give the effect on the forces and moments of an increase in the for ward speed, while the angle of attack, the elevator angle, and the throttle position re main fixed. If the coeficients of lift and drag did not change, then this would imply an increase in these forces in accordance with the speedsquared law, i.e.,
Force or moment  
(u, + A u ) ~ = 1 + 2A;
Initial force or moment u i
Since the pitching moment is initially zero, then, so long as C,,, does not change with u, it will remain zero. The situation is actually more complicated than this, for the nondimensional coefficients are in general functions of Mach number and Reynolds number, both of which increase with increasing u. The variation with Reynolds num ber is usually neglected, but the effect of Mach number must be included.
The thrust effect shows up in two different ways. One stems simply from the de rivative of thrust with speed, which depends on the type of propulsive systemjet, propeller, and so forth. The other, related mainly to propeller configurations, derives from the propulsion/airframe interaction, for example, the propeller slipstream im pinging on the wing. This is an important effect, and for some STOL airplanes, may be dominant at low speeds.
Finally, the increased loading on the airframe due to the speed increase may in duce significant structural distortion. This is a static aeroelastic effect. For example, the tail lift coefficient may be influenced appreciably by the loading (see Sec. 3.5). An appropriate variable to use for aeroelastic effects is the dynamic pressure p, =
ipv2. In order to formally include each of these three major effects, compressibility,
aeroelasticity, and propulsive, even though they would rarely all be present at the same time, each of the coefficients C,, C,, Cm is assumed to be a function of M , p,, and C, as well as angle of attack.
We then have
ac, ac, a~ ac, ap, ac, ac, c =    +  +  A ( ar ),  ( a M a; ), ( a p d a; ), (ac, ar ), (5.391)
and similarly for CZu and Cmu.
The Mach number is M = V/a, where a is the speed of sound, so
But
132 Chapter 5. The Stability Derivatives
and
and
The thrust derivative is defined in a manner consistent with Table 4.2 to be
It follows that Cxu is given by
THE DERIVATIVE C,
Since C, = T I ~ ~ V ~ S
then ac, ~ T I ~ U 2~ av  au ipv2s ipv3s au
In the reference flight condition V = uo and aVlau = 1, so
For unpowered gliding flight, T = 0 and
C, = 0 (5.337)
For constantthrust propulsion, which is an approximation for jet aircraft in cruising flight, 3 ~ 1 3 ~ = 0, and
CTU = 2C, (5.3,8)
For constantpower propulsion, which is an approximation for constantspeed pro pellers in cruising flight, TV is constant, so that
( ~ T I ~ u ) , = Toluo
and C, =  3CT0 (5.3,9)
The values of C , in the preceding expressions can be related to the reference lift and
5.3 The u Derivatives (C,,,, CzU, C,,,,,) 133
drag coefficients [see Fig. 5.1 and (5.1,1)]. Note that T, V, x are assumed to be colin ear, that is,
C, = C,, + C, sin 80 (5.3,lO)
THE DERIVATIVE CXu
From (5.1,l) we have
Since the direct aeroelastic effect on thrust is likely to be negligible, we neglect aCTlapd, and then (5.3,5) gives
When a powered windtunnel model is tested, it is common practice to measure the net axial force coefficient Cx and not its component parts C, and C,. In that case, the test data can provide the Cxu derivative directly.
THE DERIVATIVE CzU
From (5.1,1) we have that
so that
The derivative Mo(aCL/aM,) tends to be small except at transonic speeds. Theoretical values are easily calculated for highaspectratio swept wings. At subsonic speeds, the PrandtlGlauert rule combined with simple sweep theory (Kuethe and Chow, 1976) gives the lift coefficient for twodimensional flow as
aia C, = M cos A < 1
%'l  M2 cos2 A
where ai is the liftcurve slope in incompressible flow and A is the sweepback angle of the a chord line. Upon differentiation with respect to M, we get
134 Chapter 5. The Stability Derivatives
and hence
In level flight with the lift equal to the weight, MEC,, is constant, and hence Mo(aCL/aM), is proportional to 1/(1  Mg cos2 A). At supersonic speeds, the twodi mensional lift is given by Kuethe and Chow (1976)
4a cos A c, = VM' cos2 A  1
After differentiation with respect to M, we get exactly the same result as for subsonic speeds. That is (5.3,13) applies over the whole Machnumber range, except of course near M = 1 where the cited airfoil theories do not apply. Lowaspectratio wings are less sensitive to changes in M.
THE DERIVATIVE Cmu
From (5.3,5) and (5.3,6) CmU is given as
Values of aC,/aM can be found from windtunnel tests on a rigid model. They are largest at transonic speeds and are strongly dependent on the wing planform. The main factor that contributes to this derivative is the backward shift of the wing center of pressure that occurs in the transonic range. On twodimensional symmetrical wings, for example, the center of pressure moves from approximately 0 . 2 5 ~ to ap proximately 0.50~ as the Mach number increases from subsonic to supersonic values. Thus an increase in M in this range produces a divingmoment increment; that is, CmU is negative. For wings of very low aspect ratio, the center of pressure movement is much less, and the values of CmU are correspondingly smaller.
To find aC,lap, requires either an aeroelastic analysis or tests on a flexible model. As an example of this phenomenon, let us consider an airplane with a tail and a flexible fuselage.' We found in Sec. 3.5 that the tail lift coefficient is given by
The pitching moment contributed by the tail is (2.2,9)
Cm, =  V H ~ L ,
Hence
'It is not meant to imply that fuselage bending is the only important aeroelastic contribution to CmU. Distortion of the wing and tail may also be important.
5.4 The q Derivatives (Czq, Cmq) 135
When (5.3,15) is differentiated with respect topd and simplified, and the resulting ex pression is substituted into (5.3,16), we obtain the result
The corresponding contribution to CmU is [see (5.3,14)]
All the factors in this expression are positive, except for C,,,, which may be of either sign. The contribution of the tail to CmU may therefore be either positive or negative. The tail pitching moment is usually positive at high speeds and negative at low speeds. Therefore its contribution to CmU is usually negative at high speeds and posi tive at low speeds. Since the dynamic pressure occurs as a multiplying factor in (5.3,18), then the aeroelastic effect on CmU goes up with speed and down with altitude.
5.4 The q Derivatives (CZq, Cmq)
These derivatives represent the aerodynamic effects that accompany rotation of the airplane about a spanwise axis through the CG while cu, remains zero. An example of this kind of motion was treated in Sec. 3.1 (i.e., the steady pullup). Figure 5.2b shows the general case in which the flight path is arbitrary. This should be contrasted with the situation illustrated in Fig. 5.2a, where q = 0 while a, is changing.
(b)
Figure 5.2 (a) Motion with zero q, but varying cu,. (b) Motion with zero ax but varying q.
136 Chapter 5. The Stability Derivatives
Both the wing and the tail are affected by the rotation, although, when the air plane has a tail, the wing contribution to CZq and Cmq is often negligible in compari son with that of the tail. In such cases it is common practice to increase the tail effect by an arbitrary amount, of the order of lo%, to allow for the wing and body.
CONTRIBUTIONS OF A TAIL
As illustrated in Fig. 5.3, the main effect of q on the tail is to increase its angle of at tack by (ql,lu,) radians, where u, is the flight speed. It is this change in a, that ac counts for the changed forces on the tail. The assumption is implicit in the following derivations that the instantaneous forces on the tail correspond to its instantaneous angle of attack; i.e., no account is taken of the fact that it takes a finite time for the tail lift to build up to its steadystate value following a sudden change in q. (A method of including this refinement has been given by Tobak, 1954.) The derivatives obtained are therefore quasistatic.
Czq of the Tail By definition, Czy = (aCz/a~), = (2uol~)(aCzlaq),, and, from (5.1, I), (aCzlaq),
= (13C,/aq)~. The change in tail lift coefficient caused by the rotation q is
and the corresponding change in airplane lift coefficient is
' Zero lift line
Figure 5.3 Effect of pitch velocity on tail angle of attack.
5.4 The q Derivatives (CZq, Cmq) 137
Therefore
and
Cmq of the Tail The increment in pitching moment that corresponds to AC,, is [see (2.2,9)]
Hence
and
CONTRIBUTIONS OF A WING
As previously remarked, on airplanes with tails the wing contributions to the q deriv atives are frequently negligible. However, if the wing is highly swept or of low aspect ratio, it may have significant values of Czq and Cmq; and of course, on tailless air planes, the wing supplies the major contribution. The q derivatives of wings alone are therefore of great engineering importance.
Unfortunately, no simple formulas can be given, because of the complicated de pendence on the wing planform and the Mach number. However, the following dis cussion of the physical aspects of the flow indicates how linearized wing theory can be applied to the problem. Consider a plane lifting surface, at zero ax, with forward speed u, and angular velocity q about a spanwise axis (see Fig. 5.4). Each point in the wing has a velocity component, relative to the resting atmosphere, of qx normal to the surface. This velocity distribution is shown in the figure for the central and tip chords. Now there is an equivalent cambered wing that would have the identical dis tribution of velocities normal to its surface when in rectilinear translation at speed u,. This is illustrated in Fig. 5 . 5 ~ . The cross section of the curved surface S is shown in (b). The normal velocity distribution will be the same as in Fig. 5.4 if
Hence
138 Chapter 5. The Stability Derivatives
Figure 5.4 Wing velocity distribution due to pitching.
and the cross section of S is a parabolic arc. In linearized wing theory, both subsonic and supersonic, the boundary condition is the same for the original plane wing with rotation q and the equivalent curved wing in rectilinear flight. The problem of finding the q derivatives then is reduced to that of finding the pressure distribution over the equivalent cambered wing. Because of the form of ( 5 . 4 3 , the pressures are propor tional to qlu,. From the pressure distribution, C, and Cmy can be calculated. The de rivatives can in principle also be found by experiment, by testing a model of the equivalent wing.
The values obtained by this approach are quasistatic; i.e., they are steadystate values corresponding to ax = 0 and a small constant value of q. This implies that the flight path is a circle (as in Fig. 3.1), and hence that the vortex wake is not rectilinear. Now both the linearized theory and the windtunnel measurement apply to a straight wake, and to this extent are approximate. Since the values of the derivatives obtained are in the end applied to arbitrary flight paths, as in Fig. 5.2b, there is little point in correcting them for the curvature of the wake.
The error involved in the application of the quasistatic derivatives to unsteady flight is not as great as might be expected. It has been shown that, when the flight path is a sine wave, the quasistatic derivatives apply so long as the reduced frequency is small, that is,
where o is the circular frequency of the pitching oscillation. If 1 is the wavelength of the flight path, then
so that the condition k < 1 implies that the wavelength must be long compared to the chord, for example, 1 > 60C for k < 0.05.
5.4 The q Derivatives (CZq, Cmq) 139
(6)
Figure 5.5 The equivalent cambered wing.
DEPENDENCE ON h
Because the axis of rotation in Fig. 5.5, passes through the CG, the results obtained are dependent on h. The nature of this variation is found as follows. Let the axis of rotation be at A in Fig. 5.6, and let the associated lift and moment be
Now let the axis of rotation be moved to B, with the change in normal velocity distri bution shown on the figure. Since the two normal velocity distributions differ by a constant, (the upward translation qZ; Ah) the difference between the two pressure dis tributions is that associated with a flat plate at angle of attack
This angle of attack introduces a lift increment acting at the wing mean aerodynamic center of amount
140 Chapter 5. The Stability Derivatives
that is,
and
' ~orrnal velocity B (a)
(b) Figure 5.6 Effect of CG location on Czy, Cmy.
We see that CLq is linear in h, and can therefore be expressed as
CLq = 2CLa(h  h,) (5.4,ll)
where h, is the CG location at which CZq is zero. By virtue of (5.1,l) we get
CZq =  CLq = 2Cm(h  h,)
The pitching moment about the CG is
c m = c m a , + C L ( ~  haw)
so that
Equation (5.4,14) shows that Cmy is quadratic in h. We can write it without loss of generality as
where cmq is the maximum (least negative) value of Cmq and h is the CG location where it occurs (see Fig. 5.6b). The value of h is found by differentiating (5.4,14). This yields
5.5 The c i Derivatives (C ,_, C,,) 141
The linear theory of twodimensional thin wings gives for supersonic flow:
h = h = l 0
and for subsonic flow:
PITCH DAMPING OF PROPULSIVE JETS
When gases flow at high speed inside jet or rocket engines at the same time as the ve hicle is rotating in pitch or yaw, they react against the walls of the ducts with a force perpendicular to their velocity vector (the Coriolis force). This reaction can result in a pitching moment proportional to q, that is, in a contribution to Cm4, (and similarly to en,>. An analysis of this effect is given in Sec. 7.9 of Etkin (1972).
For jet airplanes in cruising flight this contribution to Cmy is usually negligible. Only at high values of C,, and when the Cmq of the rest of the airplane is small, would it be significant. On the other hand, a rocket booster at liftoff, when the speed is low, has practically zero external aerodynamic damping and the jet damping be comes very important.
5.5 The c i Derivatives (C,,, C,,)
The iu derivatives owe their existence to the fact that the pressure distribution on a wing or tail does not adjust itself instantaneously to its equilibrium value when the angle of attack is suddenly changed. The calculation of this effect, or its measure ment, involves unsteady flow. In this respect, the ff derivatives are very different from those discussed previously, which can all be determined on the basis of steadystate aerodynamics.
CONTRIBUTIONS OF A WING
Consider a wing in horizontal flight at zero a. Let it be subjected to a downward im pulse, so that it suddenly acquires a constant downward velocity component. Then, as shown in Fig. 5.7, its angle of attack undergoes a step increase. The lift then responds in a transient manner (the indicia1 response) the form of which depends on whether M is greater or less than 1. In subsonic flight, the vortices which the wing leaves be hind it can influence it at all future times, so that the steady state is approached only asymptotically. In supersonic flight, the upstream traveling disturbances move more slowly than the wing, so that it outstrips the disturbance field of the initial impulse in a finite time t , . From that time on the lift remains constant.
In order to find the lift associated with &, let us consider the motion of an airfoil with a small constant ff, but with q = 0. The motion, and the angle of attack, are
142 Chapter 5. The Stability Derivatives
I ./ Steadystate value
0  "
t 1
Figure 5.7 Lift response to step change in a. (After Tobak, NACA Rept. 1188.)
shown in Fig. 5.8. The method used follows that introduced by Tobak (1954). We as sume that the differential equation which relates CL(I) with a(f) is linear. Hence the method of superposition (the convolution integral) may be used to derive the re sponse to a linear a(t). Let the response to a unit step be A@). Then the lift coefficient at time I is (see Appendix A.3).
Since &(T) = constant, then
5.5 The ci Derivatives (C ,*., C,,) 143
Figure 5.8 Lift associated with a!
The ultimate CL response to a unitstep a input is CLcr. Let the lift defect be f ( t): that is,
A ( f ) = CLU  f ( f )
Then (5.5,l) becomes
= CLcza  S& (5 .52)
where S ( f ) = S:=, f ( f  T ) d ~ . The term S & is shown on Fig. 5.8. Now, if the idea of representing the lift by means of aerodynamic derivatives is to be valid, we must be able to write, for the motion in question,
C L ( f ) = CLUa(f ) + CL,& (5.5,3)
144 Chapter 5. The Stability Derivatives
Figure 5.9 Vector diagram of lift response to oscillatory a.
where CLe and C,, are constants. Comparing (5.5,2) and (5.5,3), we find that CL, = S(t^), a function of time. Hence, during the initial part of the motion, the derivative concept is invalid. However, for all finite wings? the area S( t ) converges to a finite value as t^ increases indefinitely. In fact, for supersonic wings, S reaches its limiting value in a finite time, as is evident from Fig. 5.7. Thus (5.5,3) is valid? with constant C,, for values of t^ greater than a certain minimum. This minimum is not large, being the time required for the wing to travel a few chord lengths. In the time range where S is constant, or differs only infinitesimally from its asymptotic value, the C,(i) curve of Fig. 5 . 8 ~ is parallel to CLa a. A similar situation exists with respect to C,.
We see from Fig. 5.8 that C,,, which is the lim  S(t^), can be positive for M = 0 t+m
and negative for larger values of M. There is a second useful approach to the ci derivatives, and that is via considera
tion of oscillating wings. This method has been widely used experimentally, and ex tensive treatments of wings in oscillatory motion are available in the l i t e ra t~re ,~ pri marily in relation to flutter problems. Because of the time lag previously noted, the amplitude and phase of the oscillatory lift will be different from the quasisteady val ues. Let us represent the periodic angle of attack and lift coefficient by the complex numbers
(Y = ffoeimt and CL = C ei" Lo (5.5941
where a. is the amplitude (real) of a , and C,, is a complex number such that I c ~ I is the amplitude of the CL response, and arg C,, is its phase angle. The relation between Ch and a, appropriate to the low frequencies characteristic of dynamic stability is il lustrated in Fig. 5.9. In terms of these vectors, we may derive the value of CL, as fol lows. The ci vector is
Thus C, may be expressed as
3For twodimensional incompressible flow, the area S( i ) diverges as r + m. That is, the derivative concept is definitely not applicable to that case.
4Exactly for supersonic wings, and approximately for subsonic wings.
'See bibliography.
Hence
5.5 The dr Derivatives (C,_, C,,) 145
Figure 5.10 Lift on oscillating twodimensional airfoil.
or, if the amplitude a, is unity, C,, = I[C,]lk, where k is the reduced frequency ~ / 2 u o .
To assist in forming a physical picture of the behavior of a wing under these con ditions, we give here the results for a twodimen~ional,~ airfoil in incompressible flow. The motion of the airfoil is a plunging oscillation; that is, it is like that shown in Fig. 5.2a, except that the flight path is a sine wave. The instantaneous lift on the air foil is given in two parts (see Fig. 5.10):
where
and F(k) and G(k) are the real and imaginary parts of the Theodorsen function C(k) plotted in Fig. 5.11 (Theodorsen, 1934). The lift that acts at the midchord is propor tional to iu = zlu,, where z is the translation (vertically downward) of the airfoil. That is, it represents a force opposing the downward acceleration of the airfoil. This force is exactly that which is required to impart an acceleration z to a mass of air contained in a cylinder, the diameter of which equals the chord c. This is known as the "appar ent additional mass." It is as though the mass of the airfoil were increased by this amount. Except in cases of very low relative density p = 2mlpSF, this added mass is small compared to that of the airplane itself, and hence the force C,, is relatively unimportant. Physically, the origin of this force is in the reaction of the air which is associated with its downward acceleration. The other component, CL,, which acts at the chord point, is associated with the circulation around the airfoil, and is a conse quence of the imposition of the KuttaJoukowski condition at the trailing edge. It is seen that it contains one term proportional to a and another proportional to iu. From Fig. 5.10, the pitchingmoment coefficient about the CG is obtained as
'Rodden and Giesing (1970) have extended and generalized this method. In particular they give re sults for finite wings.
146 Chapter 5. The Stability Derivatives
Reduced frequency, k
Reduced frequency, k
Figure 5.11 The Theodorsen function.
From [(5.5,6) and (5.5,7)], the following derivatives are found for frequency k.
The awkward situation is evident, from (5.5.8), that the derivatives are frequencyde pendent. That is, in free oscillations one does not know the value of the derivative un til the solution to the motion (i.e. the frequency) is known. In cases of forced oscilla tions at a given frequency, this difficulty is not present.
When dealing with the rigidbody motions of flight vehicles, the characteristic nondimensional frequencies k are usually small, k 4 1. Hence it is reasonable to use the F(k) and G(k) corresponding to k += 0. For the twodimensional incompressible case described above, lim F(k) = 1, so that CLa = 27r and Cma = 27T(h  t), the theo retical steadyflow valik; This conclusion, that CLa and Cmn are the quasistatic values, also holds for finite wings at all Mach numbers. The results for CLa and C,, are not so clear, however, since lim G(k)lk given above is infinite. This singularity is marked for the example of twodimensional flow given above, but is not evident for finite wings at moderate aspect ratio. Miles (1950) indicates that the k log k term responsi
5.5 The dv Derivatives (CLU, C,,) 147
ble for the singularity is not significant for aspect ratios less than 10, and the numeri cal calculations of Rodden and Giesing (1970) show no difficulty at values of k as low as 0.001. Filotas' (1971) solutions for finite wings bear out Miles' contention. Thus for finite wings definite values of C,, and C,, can be associated with small but nonvanishing values of k. The limiting values described above can be obtained from a firstorderinfrequency analysis of an oscillating wing. To summarize, the c i deriva tives of a wing alone may be computed from the indicia1 response of lift and pitching moment, or from firstorderinfrequency analysis of harmonically plunging wings.
CONTRIBUTIONS OF A TAIL
There is an approximate method for evaluating the contributions of a tail surface, which is satisfactory in many cases. This is based on the concept of the lag of the downwash. It neglects entirely the nonstationary character of the lift response of the tail to changes in tail angle of attack, and attributes the result entirely to the fact that the downwash at the tail does not respond instantaneously to changes in wing angle of attack. The downwash is assumed to be dependent primarily on the strength of the wing's trailing vortices in the neighborhood of the tail. Since the vorticity is con vected with the stream, then a change in the circulation at the wing will not be felt as a change in downwash at the tail until a time At = l,lu, has elapsed, where 1, is the tail length. It is therefore assumed that the instantaneous downwash at the tail, ~ ( t ) , corresponds to the wing a at time ( t  At). The corrections to the quasistatic down wash and tail angle of attack are therefore
C,, of a Tail The correction to the tail lift coefficient for the downwash lag is
The correction to the airplane lift is therefore
Therefore
and
148 Chapter 5. The Stability Derivatives
C,, of a Tail
The correction to the pitching moment is obtained from AC,, as
Therefore
and
5.6 The P Derivatives (C,,, C,,, C,,)
These derivatives all are obtainable from windtunnel tests on yawed models (Camp bell and McKinney, 1952). Generally speaking, estimation methods do not give com pletely reliable results, and testing is a necessity.
THE DERIVATIVE C,,
This is the sideforce derivative, giving the force that acts in the y direction (right) when the airplane has a positive /3 or v (i.e., a sideslip to the right, see Fig. 3.1 1). C,, is usually negative, and frequently small enough to be neglected entirely. The main contributions are those of the body and the vertical tail, although the wing, and wing body interference, may modify it significantly. Of these, only the tail effect is readily estimated. It may be expressed in terms of the verticaltail liftcurve slope and the sidewash factor (see Sec. 3.9). (In this and the following sections the fin velocity ra tio VF/V is assumed to be unity.)
The most troublesome component of this equation is the sidewash derivative aalap, which is difficult to estimate because of its dependence on the wing and fuselage geometry (see Sec. 3.9).
THE DERIVATIVE C,,
C,, is the dihedral effect, which was discussed at some length in Sec. 3.12.
5.7 The p Derivatives (Cyp, C6, CnP) 149
THE DERIVATIVE C,,
C,, is the weathercock stability derivative, dealt with in Sec. 3.9.
5.7 The p Derivatives (C,,, Clp, CnP )
When an airplane rolls with angular velocity p about its x axis (the flight direction), its motion is instantaneously like that of a screw. This motion affects the airflow (lo cal angle of attack) at all stations of the wing and tail surfaces. This is illustrated in Fig. 5.12 for two points: a wing tip and the fin tip. It should be noted that the nondi mensional rate of roll, j j = pb/2u0 is, for small p, the angle (in radians) of the helix traced by the wing tip. These angle of attack changes bring about alterations in the aerodynamic load distribution over the surfaces, and thereby introduce perturbations in the forces and moments. The change in the wing load distribution also causes a modification to the trailing vortex sheet. The vorticity distribution in it is no longer symmetrical about the x axis, and a sidewash (positive, i.e., to the right) is induced at a vertical tail conventionally placed. This further modifies the angleofattack distri bution on the verticaltail surface. This sidewash due to rolling is characterized by the derivative aa/ai. It has been studied theoretically and experimentally by Michael (1952), who has shown its importance in relation to correct estimation of the tail con tributions to the rolling derivatives. Finally, the helical motion of the wing produces a trailing vortex sheet that is not flat, but helical. For the small rates of roll admissible in a linear theory, this effect may be neglected with respect to both wing and tail forces.
THE DERIVATIVE CYp
The side force due to rolling is often negligible. When it is not, the contributions that need to be considered are those from the wing7 and from the vertical tail. The verti caltail effect may be estimated in the light of its angleofattack change (see Fig. 5.12) as follows. Let the mean change in a, (see Fig. 3.12) due to the rolling velocity be
where 2, is an appropriate mean height of the fin. Introducing the nondimensional rate of roll, we may rewrite this as
The incremental sideforce coefficient on the fin is obtained from Aa,,
AC,, = a , ha, = a,jj 2    ( 2 :;) 'For the effect of the wing at low speeds, see Campbell and McKinney (1952).
150 Chapter 5. The Stability Derivatives
Figure 5.12 Angle of attack changes due top.
where a, is the liftcurve slope of the vertical tail. The incremental side force on the airplane is then given by
thus
THE DERIVATIVE ClP
C,p is known as the dampinginroll derivative. It expresses the resistance of the air plane to rolling. Except in unusual circumstances, only the wing contributes signifi cantly to this derivative. As can be seen from Fig. 5.12, the angle of attack due to p varies linearly across the span, from the value pb/2u, at the right wing tip to pb/2uo at the left tip. This antisymmetric a distribution produces an antisymmetric incre ment in the lift distribution as shown in Fig. 5.13. In the linear range this is superim posed on the symmetric lift distribution associated with the wing angle of attack in undisturbed flight. The large rolling moment L produced by this lift distribution is proportional to the tip angle of attack j? (see Fig. 5.12), and ClP is a negative constant, so long as the local angle of attack remains below the local stalling angle.
If the wing angle of attack at the center line, a,(O), is large, then the incremental value due to p may take some sections of the wing beyond the stalling angle, as
5.7 The p Derivatives (CJp, ClP, CnP) 151
I
Figure 5.13 Spanwise lift distribution due to rolling.
shown in Fig. 5.14. [Actually, for finite span wings, there is an additional induced an gle of attack distribution a,(y) due to the vortex wake that modifies the net sectional value still further. We neglect this correction here in the interest of making the main point.] When this happens IC,~P~ is reduced in magnitude from the linear value and if a,(O) is large enough, will even change sign. When this happens, the wing will au torotate, the main characteristic of spinning flight.
THE DERIVATIVE Cap
The yawing moment produced by the rolling motion is one of the socalled cross de rivatives. It is the existence of these cross derivatives that causes the rolling and yaw ing motions to be so closely coupled. The wing and tail both contribute to C,,,.
The wing contribution is in two parts. The first comes from the change in profile drag associated with the change in wing angle of attack. The wing a is increased on the righthand side and decreased on the lefthand side. These changes will normally be accompanied by an increase in profile drag on the right side, and a decrease on the
'C 0
'J E 8 !E   8  M
g v t a l l e d portion of wing
0 ad$) aw(~) ad)) * Net section angle of attack
Figure 5.14 Reduction of C,p due to wing stall.
152 Chapter 5. The Stability Derivatives
Y
Figure 5.15 Inclination of C, vector due to rolling.
left side, combining to produce a positive (noseright) yawing moment. The second wing effect is associated with the foreandaft inclination of the lift vector caused by the rolling in subsonic flight and in supersonic flight when the leading edge is sub sonic. It depends on the leadingedge suction. The physical situation is illustrated in Fig. 5.15. The directions of motion of two typical wing elements are shown inclined by the angles ? 6 = pylu, from the direction of the vector u,. Since the local lift is perpendicular to the local relative wind, then the lift vector on the right half of the wing is inclined forward, and that on the left half backward. The result is a negative yawing couple, proportional to the product C,B. If the wing leading edges are super sonic, then the leadingedge suction is not present, and the local force remains nor mal to the surface. The increased angle of attack on the right side causes an increase in this normal force there, while the opposite happens on the left side. The result is a positive yawing couple proportional to j3.
The tail contribution to Cnp is easily found from the tail side force given previ ously (5.7,2). The incremental C, is given by
where I F is the distance shown in Fig. 3.12. Therefore
and
where Vv is the verticaltail volume ratio.
5.8 The r Derivatives (CYr, CIr, C,,) 153
5.8 The r Derivatives (Cyr, C, , Cnr)
When an airplane has a rate of yaw r superimposed on the foward motion u,, its ve locity field is altered significantly. This is illustrated for the wing and vertical tail in Fig. 5.16. The situation on the wing is clearly very complicated when it has much sweepback. The main feature however, is that the velocity of the chord line normal to itself is increased by the yawing on the lefthand side, and decreased on the right side. The aerodynamic forces at each section (lift, drag, moment) are therefore in creased on the lefthand side, and decreased on the righthand side. As in the case of the rolling wing, the unsymmetrical lift distribution leads to an unsymmetrical trail ing vortex sheet, and hence a sidewash at the tail. The incremental tail angle of attack is then
rl, a ( ~ A a , =  + r 
u, d r
Figure 5.16 Velocity field due to yawing. A% = velocity vector due to rate of yaw r.
154 Chapter 5. The Stability Derivatives
THE DERIVATIVE C,,,
The only contribution to Cyr that is normally important is that of the tail. From the an gle of attack change we find the incremental C, to be
thus
THE DERIVATIVE C,
This is another important cross derivative; the rolling moment due to yawing. The in crease in lift on the left wing, and the decrease on the right wing combine to produce a positive rolling moment proportional to the original lift coefficient C,. Hence this derivative is largest at low speed. Aspect ratio, taper ratio, and sweepback are all im portant parameters.
When the vertical tail is large, its contribution may be significant. A formula for it can be derived in the same way as for the previous tail contributions, with the result
THE DERIVATIVE Cnr
Cnr is the dampinginyaw derivative, and is always negative. The body adds a negli gible amount to Cnr except when it is very large. The important contributions are those of the wing and tail. The increases in both the profile and induced drag on the left wing and the decreases on the right wing give a negative yawing moment and hence a resistance to the motion. The magnitude of the effect depends on the aspect ratio, taper ratio, and sweepback. For extremely large sweepback, of the order of 60°, the yawing moment associated with the induced drag may be positive; that is, pro duce a reduction in the damping.
The side force on the tail also provides a negative yawing moment. The calcula tion is similar to that for the preceding tail contributions, with the result
5.9 Summary of the Formulas
The formulas that are frequently wanted for reference are collected in Tables 5.1 and 5.2. Where an entry in the table shows only a tail contribution, it is not implied that the wing and body effects are not important, but only that no convenient formula is available.
6
uU
,.
0'1 i m m
h" i, +
i,? 1 2 m "
"4,
+ GI E m m
E"
GI u" m m
h" i, '
G I P m m
djl m m
s7 I
,. u"
m m I  v
c;; +
;.lz m m "4,
1 0 "
i,q2 m
I
u"1r m m w
9
;;
I
g g ; . 2
&
i" +
a s N ' I *
g
a
2 I2
? I *
212 
I
I Z
.a
, ,u z
% N I *
5 N
I J
$ 9 Z
~W
156 Chapter 5. The Stability Derivatives
Table 5.2 SummaryLateral Derivatives
*means contribution of the tail only, formula for wingbody not available; V,/V = 1.
N.A. means no formula available.
CY
5.1 0 Aeroelastic Derivatives
In Sec. 4.12 there were introduced aerodynamic derivatives associated with the defor mations of the airplane. These are of two kinds: those that appear in the rigidbody equations and those that appear in the added equations of the elastic degrees of free dom. These are illustrated in this section by consideration of the hypothetical vibra tion mode shown in Fig. 5.17. In this mode it is assumed that the fuselage and tail are rigid, and have a motion of vertical translation only. The flexibility is all in the wing, and it bends without twisting. The functions describing the mode (4.11,l) are there fore:
N.A.
N.A.
S, zF 1, au *Y
c,
For the generalized coordinate, we have used the wingtip deflection 2,. h(Y) is then a normalized function describing the wing bending mode.
Since the elastic degrees of freedom are only important in relation to stability and control when their frequencies are relatively low, approaching those of the rigid body modes, then it is reasonable to use the same approximation for the aerodynamic forces as is used in calculating stability derivatives. That is, if quasisteady flow the ory is adequate for the aerodynamic forces associated with the rigidbody motions, then we may use the same theory for the elastic motions.
C"
I C.G. of deformed air~lane
Figure 5.17 Symmetrical wing bending.
5.10 Aeroelustic Derivatives 157
In the example chosen, we assume that the only significant forces are those on the wing and tail, and that these are to be computed from quasisteady flow theory. In the light of these assumptions, some of the representative derivatives of both types are discussed below. As a preliminary, the forces induced on the wing and tail by the elastic motion are treated first.
FORCES ON THE WING
The vertical velocity of the wing section distant y from the center line is
i = h(Y)iT (5.10,2)
and the corresponding change in wing angle of attack is
A a b ) = h(Y)iTIuO
This angle of attack distribution can be used with any applicable steadyflow wing theory to calculate the incremental local section lift. (It will of course be proportional to iJu,.) Let it be denoted in coefficient form by C;(Y) iT/~O, and the corresponding increment in wing total lift coefficient by C;,iT/uO Ci(Y) and CLw are thus the values corresponding to unit value of the nondimensional quantity iTluO.
FORCE ON THE TAIL
The tail experiences a downward velocity h(0)iT, and also, because of the altered wing lift distribution, a downwash change (ae/diT)iT. Hence the net change in tail an gle of attack is
a€ Aa, = h(O)i,/u,   iT
aiT
This produces an increment in the tail lift coefficient of amount
THE DERIVATIVE Zi,
This derivative describes the contribution of wing bending velocity to the Z force act ing on the airplane. A suitable nondimensional form is aC,l~(i,lu,). Since C, = C,, we have that
and hence
158 Chapter 5. The Stability Derivatives
THE DERIVATIVE AnW
This derivative [see (4.12,12)] represents the contribution to the generalized force in the bending degree of freedom, associated with a change in the w velocity of the air plane. A suitable nondimensional form is obtained by defining
and using ax in place of w (w = u,cu,). Then the appropriate nondimensional deriva tive is C,o.
Let the wing lift distribution due to a perturbation a in the angle of attack (con stant across the span) be given by CIaQ)a. Then in a virtual displacement in the wing bending mode $ , the work done by this wing loading is
where c b ) is the local wing chord. The corresponding contribution to 9 is
and to C, is
The tail also contributes to this derivative, for the tail lift associated with a is
and the work done by this force during the virtual displacement is
ata ( 1   I):  pu;S,h(O) Sz,
Therefore the contribution to C, is
and to Cga is
The total value of Cgo is then the sum of 5.10,6 and 5.10,7.
THE DERIVATIVE b,, (see 4.12,12)
This derivative identifies the contribution of 2, to the generalized aerodynamic force in the distortion degree of freedom. We have defined the associated wing load distri
5.11 Exercises 159
bution above by the local lift coefficient C;(y)i,lu,. As in the case of the derivative An, above, the work done by this loading is calculated, with the result that the wing contributes
Likewise, the contribution of the tail is calculated here as for A,,, and is found to be
The total value of aC,la(i,lu,) is then the sum of 5.10,8 and 5.10,9.
5.11 Exercises
(2); Estimate the magnitude of this term 5.1 The derivative CzL, contains the term M, 
for an airplane with wing loading 70 psf (3,352 Pa) flying at 20,000 ft (6,096 m) alti tude, for Mach numbers between 0.2 and 0.8. The following data pertain to the wing:
Sweep (a chord) A = 30"
S = 5,500 ft2 (5 11.0 m2)
Plot the result vs. M,. Calculate the contribution this term makes to Z, and plot this as well. (Compare with Z, for the B747 from Table 6.2, and comment).
5.2 A windtunnel model is mounted with one degree of freedompivoted so that it can only rotate about the yaxis of the body frame, which is perpendicular to the relative wind. It is elastically restrained with a pitching moment M = kO. Show how the sum (Cmc, + Cma) can be estimated from experiments in which the model is free to os cillate in pitch with wind on and off. Assume M, can be neglected with the wind off and Z, and ZG can be neglected.
5.3 Consider the windlfin system of Fig. 5.16, with the following properties:
Wing: A = 5; h = 0.5; A,, = 30"; r variable
Fin: a, = 3.5 radI; 1,lb = 0.5; zFlb = 0.1; VV variable; aulap negligible.
Estimate values of the stability derivatives (for haw, = h and L/D = 12)
at C,, = 1 .O. Plot the spiral stability boundary for horizontal flight:
E = C,,Cnr  ClrCnO = 0
[see (6.8,6) with O,, = 01 in the plane of V , vs. I'. (Make any reasonable assumptions you need to supplement the given data).
5.4 A jet airplane has a thrust line that passes above the CG by a distance equal to 10% of the M.A.C. With the assumption aTlau = 0, estimate the increment thus caused in
~ m , ;
160 Chapter 5. The Stability Derivatives
5.5 Find Cnp due to the tilting of the lift vector for a wing with an elliptic lift distribution l2 4y2
i.e., a wing with lift per unit span 16) which obeys  +  = 1 . Assume that the a2 b2
tilt angle is small. Express Cnp in terms of CL, the lift coefficient of the wing when it is not rolling.
5.6 Assume that Figs. 5.7 and 5.8 are experimental measurements. Select an analytic function CLstep(t) that can represent Fig. 5.7 (M = 0 case). Find the corresponding transfer function relating CL to a. Use this transfer function to generate a function of time corresponding to Fig. 5.8b and demonstrate that it has the desired form.
5.12 Additional Symbols Introduced in Chapter 5
e efficiency factor for winginduced drag
k reduced frequency, wd2u,
p, dynamic pressure, i p V 2
w circular frequency
See also Secs. 2.11,3.15, and 4.14.
C H A P T E R 6
Stability of Uncontrolled Motion
The preceding chapters have been to some extent simply the preparation for what fol lows in this and the succeeding two chapters, that is, a treatment of the uncontrolled and controlled motions of an airplane. The system model was developed in Chap. 4, and the aerodynamic ingredients were described in Chaps. 2, 3, and 5. In this chapter we tackle first the simplest of these cases, the uncontrolled motion, that is, the motion when all the controls are locked in position. An airplane in steady flight may be sub jected to a momentary disturbance by a nonuniform or nonstationary atmosphere, or by movements of passengers, release of stores, and so forth. In this circumstance some of the questions to be answered are, "What is the character of the motion fol lowing the cessation of the disturbance? Does it subside or increase? If it subsides what is the final flight path?'The stability of small disturbances from steady flight is an extremely important property of aircraftfirst, because steady flight conditions make up most of the flight time of airplanes, and second, because the disturbances in this condition must be small for a satisfactory vehicle. If they were not it would be unacceptable for either commercial or military use. The required dynamic behavior is ensured by designby making the smalldisturbance properties of concern (the nat ural modes) such that either human or automatic control can keep the disturbances to an acceptably small level. Finally the smalldisturbance model is actually valid for disturbance magnitudes that seem quite violent to human occupants.
6.1 Form of Solution of SmallDisturbance Equations
The smalldisturbance equations are (4.9,18 and 4.9,19). They are both of the form
x = Ax + Af, (6.1,l)
where x is the (N X 1) state vectol; A is the (N X N ) system matrix, a constant, and Af,. is the (N X 1) vector of incremental control forces and moments. In this applica tion, the control force vector is zero, so the equation to be studied is
x = Ax (6.1,2)
Solutions of this firstorder differential equation are well known. They are of the form
x(t) = x,eAt (6.1,3)
x,, is an eigenvector and A is an eigenvalue of the system. x, is also seen to be the value of the state vector at t = 0. Substitution of (6.1,3) into (6.1,2) gives
162 Chapter 6. Stability of Uncontrolled Motion
Ax, = Ax,
(A  AI)x, = 0
where I is the identity matrix. Since the scalar expansion of (6.1,5) is a system of N homogeneous equations (zeros on the righthand side) then there is a nonzero solu tion for x, only when the system determinant vanishes, that is, when
det (A  AI) = 0 (6.1,6)
The determinant in (6.1,6) is the characteristic determinant of the system. When ex panded, the result is a polynomial in A of degree N, the characteristic polynomial, and the Nth degree algebraic equation (6.1,6) is the characteristic equation of the system. Since the equation is of the Nth degree it has in general N roots A, some real and some occurring in conjugate complex pairs. Corresponding to each real eigen value A is a real eigenvector x,, and to each complex pair hi and A; there corresponds a conjugate complex pair of eigenvectors x, and x:. Since any one of the A's can pro vide a solution to (6.1,2) and since the equation is linear, the most general solution is a sum of all the corresponding x(t) of (6.1,3), that is,
Each of the solutions described by (6.1,3) is called a natural mode, and the general solution (6.1,7) is a sum of all the modes. A typical variable, say w, would, according to (6.1,7) have the form
where the ai would be fixed by the initial conditions. The pair of terms corresponding to a conjugate pair of eigenvalues
Upon expanding the exponentials, (6.1,10) becomes
en'(Al cos wt + A, sin wt) (6.1, l l )
where A, = (a , + a,) and A, = i(a,  a,) are always real. That is, (6.1,11) describes an oscillatory mode, of period T = 2 d o , that either grows or decays, depending on the sign of n. The four kinds of mode that can occur, according to whether A is real or complex, and according to the sign of n are illustrated in Fig. 6.1. The disturbances shown in (a) and (c ) increase with time, and hence these are unstable modes. It is conventional to refer to (a) as a static instability or divergence, since there is no ten dency for the disturbance to diminish. By contrast, ( c ) is called dynamic instability or a divergent oscillation, since the disturbance quantity alternately increases and di minishes, the amplitude growing with time. (b) illustrates a subsidence or conver gence, and (6) a damped or convergent oscillation. Since in both (b ) and (6) the dis turbance quantity ultimately vanishes, they represent stable modes.
It is seen that a "yes" or "no" evaluation of the stability is obtained simply from the signs of the real parts of the As. If there are no positive real parts, there is no in stability. This information is not sufficient, however, to evaluate the handling quali
6.1 Form of Solution of SmallDisturbance Equations 163

half
Figure 6.1 Types of solution. (a ) A real, positive. (b) A real, negative. (c ) A complex, n > 0. (d) A complex, n < 0.
ties of an airplane (see Chap. 1). These are dependent on the quantitative as well as on the qualitative characteristics of the modes. The numerical parameters of primary interest are
2% 1. Period. T = 
2. Time to double or time to half.
3. Cycles to double (N,,,,,,) or cycles to half (N,,,,,).
The first two of these are illustrated in Fig. 6.1. When the roots are real, there is of course no period, and the only parameter is the time to double or half. These are the times that must elapse during which any disturbance quantity will double or halve it self, respectively. When the modes are oscillatory, it is the envelope ordinate that doubles or halves. Since the envelope may be regarded as an amplitude modulation,
164 Chapter 6. Stability of Uncontrolled Motion
then we may think of the doubling or halving as applied to the variable amplitude. By noting that log, 2 = log, $ = 0.693, the reader will easily verify the following rela tions:
Time to double or half:
Cycles to double o r half:
Logarithmic decrement (log of ratio of successive peaks):
In the preceding equations,
wn = (w2 + n2)'I2, the "undamped" circular frequency 5   nlw,, the damping ratio
COMPUTATION OF EIGENVALUES AND EIGENVECTORS
As noted above, the eigenvalues and eigenvectors are properties of the matrix A. A number of software packages such as MATLAB are now available for calculating them. For all the numerical examples in this book we have used the Student Version of Program CC.' Appendix A.5 shows how we used it to get the results.
ROUTH'S CRITERIA FOR STABILITY
The stability of the airplane is governed, as we have seen, by the real parts of the eigenvalues, the roots of the characteristic equation. Now it is not necessary actually to solve the characteristic equation (6.1,6) for these roots in order to discover whether there are any unstable ones. E. J. Routh (1905) has derived a criterion that can be ap plied to the coefficients of the equation to get the desired result. The criterion is that a certain set of test functions shall all be positive (Etkin, 1972). We present below the result for the important case of the quartic equation, which will turn up later in this chapter.
Let the quartic equation be
Then the test functions are F,, = A, F, = B, F2 = BC  AD, F3 = F2D  B'E, F4 =
'Available from Systems Technology Inc. Hawthorne, CA.
6.2 Longitudinal Modes of a Jet Transport 165
F,BE. The necessary and sufficient conditions for these test functions to be positive are
A, B, D, E > 0
and
R = D(BC  AD)  B2E > 0 (6.1,14)
It follows that C also must be positive. The quantity on the lefthand side of (6.1,14) is commonly known as Routh 's discriminant.
Duncan (1952) has shown that the vanishing of E and of R represent significant critical cases. If the airplane is stable, and some design parameter is then varied in such a way as to lead to instability, then the following conditions hold:
1. If only E changes from + to , then one real root changes from negative to positive; that is, one divergence appears in the solution (see Fig. 6. la).
2. If only R changes from + to , then the real part of one complex pair of roots changes from negative to positive; that is, one divergent oscillation appears in the solution (see Fig. 6. lc).
Thus the conditions E = 0 and R = 0 define boundaries between stability and in stability. The former is the boundary between stability and static instability, and the latter is the boundary between stability and a divergent oscillation. These stability boundaries are very useful for certain analytical purposes. We shall in particular make use of the E = 0 boundary.
6.2 Longitudinal Modes of a Jet Transport
The foregoing theory is now illustrated by applying it to the Boeing 747 transport. The needed geometrical and aerodynamic data for this airplane are given in Appendix E. The flight condition for this example is cruising in horizontal flight at approxi mately 40,000 ft at Mach number 0.8. Relevant data are as follows:
W = 636,636 lb (2.83176 X lo6 N) S = 5500 ft2 (5 1 1.0 m2)
i? = 27.31 ft (8.324 m) b = 195.7 ft (59.64 m)
I, = 0.183 X 10' slug ft2 (0.247 X lo8 kg m2) I, = 0.33 1 X lo8 slug ft2 (0.449 X 10' kg m2)
I , = 0.497 X lo8 slug ft2 (0.673 X lo8 kg m2) I, =  .I56 X 10' slug ft2 (.212 X lo7 kg m2)
u, = 774 fps (235.9 mls) 6, = 0 p = 0.0005909 slug/ft3 (0.3045 kg/m3)
C,, = 0.654 C,,, = 0.
The preceding four inertias are for stability axes at the stated flight condition. In the numerical examples of this and the following two chapters, the system matrices and the solutions are all given in English units. The nondimensional stability derivatives
166 Chapter 6. Stability of Uncontrolled Motion
Table 6.1 Nondimensional DerivativesB747 Airplane
c x cz c m
ii 0.1080 0.1060 0.1043 a 0.2193 4.920  1.023 4 0 5.921 23.92 6 0 5.896 6.314
are given in Table 6.1, and the dimensional derivatives in Table 6.2. With the above data we calculate the system matrix A for this case. (Recall that the state vector is [Au w q A0lT).
0.006868 0.01395
A = [ 0.09055 0.315 1 773.98 0.0001 187 0.001026 0.4285 0
0 0 1 0
The characteristic equation (6.1,6) is next calculated to be:
The two stability criteria are
and
so that there are no unstable modes.
EIGENVALUES
The roots of the characteristic equation (6.2,2), the eigenvalues, are
Mode 1 (Phugoid mode): A,,, = 0.003289 2 0.067231'
Mode 2 (Shortperiod mode): A , , = 0.3719 + 0.88751' ( 6 . 2 3
We see that the natural modes are two damped oscillations, one of long period and lightly damped, the other of short period and heavily damped. This result is quite typ
Table 6.2 Dimensional DerivativeeB747 Airplane
6.2 Longitudinal Modes of a Jet Transport 167
Table 6.3
Period thdf Nhay
Mode Name (s) ( s ) (cycles)
"The phugoid mode was first described by Lanchester (l908), who also named it. The name comes from the Greek root forpee as infugitive. Actually Lanchester wanted the root for fly. Appropriate or not, the word phugoid has become established in aeronautical jargon.
ical. The modes are conventionally named as in Table 6.3, which also gives their peri ods and damping. The transient behavior of the state variables in these two modes is displayed in Fig. 6.2.
EIGENVECTORS
The eigenvectors corresponding to the above modes are given in Table 6.4. They are for the nondimensionable variables, in polar form, the values given corresponding to n + iw. Eigenvectors are arbitrary to within a complex factor, so it is only the relative values of the state variables that are significant. We have therefore factored them to
8 747 Phugo~d mode
B 747 Shortperiod mode
T~me, s
(b )
Figure 6.2 Characteristic transients. (a ) Phugoid mode. (b) Shortperiod (pitching) mode.
168 Chapter 6. Stability of Uncontrolled Motion
Table 6.4 Eigenvectors (polar form)
Phugoid ShortPeriod
Magnitude Phase Magnitude Phase
make A0 equal to unity, as displayed in the Argand diagram of Fig. 6.3. As we have chosen the positive value of w, the diagrams can be imagined to be rotating counter clockwise and shrinking, with their projections on the real axis being the real values of the variables.
The phugoid is seen to be a motion in which the pitch rate 4 and the angle of at tack change a are very small, but Ali and A0 are present with significant magnitude. The speed leads A0 by about 90" in phase.
The shortperiod mode, by contrast, is one in which there is negligible speed variation, while the angle of attack oscillates with an amplitude and phase not much different from that of AO. This mode behaves like one with only two degrees of free dom, A0 and a.
FLIGHT PATHS IN THE CHARACTERISTIC MODES
Additional insight into the modes is gained by studying the flight path. With the at mosphere at rest, the differential equations for the position of the CG in FE are given by (4.9,10), with 0, = 0, that is,
In a characteristic oscillatory mode with eigenvalues A, A*, the variations of Au, 0, and w are [cf. (6.1,8)]
* A*' Au = qjeA' + u ,je * A*' w = + uzje (6.2,5) * A*' 0 = u4;eAt + u4;e
where the constants uy are the components of the eigenvector corresponding to A. For the previous numerical example, they are the complex numbers given in polar form in Table 6.4. After substituting (6.2,5) in (6.2,4) and integrating from t = 0 to t we get
[%j Au~u4j eiw zE = 2enf Re I + const
6.2 Longitudinal Modes of a Jet Transport 169
a = 2 = 0.036 q^ = 0.001 2 \ (not visible)
$ = 0.017 A$ = 0.029 (not visible) (not vls~ble)
(b)
Figure 6.3 (a ) Vector diagram of phugoid mode. (b) Vector diagram of shortperiod mode.
where Re denotes the real part of the complex number in the square brackets. For the numerical data of the above example (6.2,6) has been used to calculate the flight paths in the two modes, plotted in Fig. 6.4. The nonzero initial conditions are arbi trary, and the trajectories for both modes asymptote to the steady reference flight path. Figure 6.4 shows that the phugoid is an undulating flight of very long wave length. The mode diagram, Fig. 6 . 3 ~ shows that the speed leads the pitch angle by about 90°, from which we can infer that u is largest near the bottom of the wave and least near the top. This variation in speed results in different distances being traversed
170 Chapter 6. Stability of Uncontrolled Motion
1,000 1 I I 2,000 4,000 6.000
x, ft
(c)
Figure 6.4 (a) Phugoid flight path (fixed reference frame). (b) Phugoid flight path (moving reference frame). (c ) Shortperiod flight path.
during the upper and lower halves of the cycle, as shown in Fig. 6.4~ . For larger am plitude oscillations, this lack of symmetry in the oscillation becomes much more pro nounced (although the linear theory then fails to describe it accurately) until ulti mately the upper part becomes first a cusp and then a loop (see Miele, 1962, p. 273). The motion (see Sec. 6.3) is approximately one of constant total energy, the rising
6.3 Approximate Equations for the Longitudinal Modes 171
and falling corresponding to an exchange between kinetic and potential energy. Fig ure 6.4b shows the phugoid motion relative to axes moving at the reference speed u,. This is the relative path that would be seen by an observer flying alongside at speed 4.
Figure 6 . 4 ~ shows the path for the shortperiod mode. The disturbance is rapidly damped. The transient has virtually disappeared within 3000 ft of flight, even though the initial A a and A 8 were very large. The deviation of the path from a straight line is small, the principal feature of the motion being the rapid rotation in pitch.
6.3 Approximate Equations for the Longitudinal Modes
The numerical solutions for the modes, although they certainly show their properties, do not give much physical insight into their genesis. Now each oscillatory mode is equivalent to some secondorder massspringdamper system, and each nonoscilla tory mode is equivalent to some massdamper system. To understand the modes, and the influence on them of the main flight and vehicle parameters, it is helpful to know what contributes to the equivalent masses, springs, and dampers. To achieve this re quires analytical solutions, which are simply not available for the full system of equations. Hence we are interested in getting approximate analytical solutions, if they can reasonably represent the modes. Additionally, approximate models of the in dividual modes are frequently useful in the design of automatic flight control systems (McRuer et al., 1973). In the following we present some such approximations and the methods of arriving at them.
There are two approaches generally used to arrive at these approximations. One is to write out a literal expression for the characteristic equation and, by studying the order of magnitude of the terms in it, to arrive at approximate linear or quadratic fac tors. For example, if the characteristic equation (6.1,13) is known to have a "small" real root, an approximation to it may be obtained by neglecting all the higher powers of A, that is,
D A + E = O
Or if there is a "large" complex root, it may be approximated by keeping only the first three terms, that is,
A A ~ + B A + C = O
This method is frequently useful, and is sometimes the only reasonable way to get an approximation.
The second method, which has the advantage of providing more physical insight, proceeds from a foreknowledge of the modal characteristics to arrive at approximate system equations of lower order than the exact ones. For the longitudinal modes we use the second method (see below), and for the lateral modes (see Sec. 6.8) both methods are needed.
It should be noted that no simple analytical approximations can be relied on to give accurate results under all circumstances. Machine solutions of the exact matrix is the only certain way. The value of the approximations is indicated by examples in the following.
To proceed now to the phugoid and shortperiod modes, we saw in Fig. 6.3 that some state variables are negligibly small in each of the two modes. This fact suggests
172 Chapter 6. Stability of Uncontrolled Motion
certain approximations to them based on reduced sets of equations of motion arrived at by physical reasoning. These approximations, which are quite useful, are devel oped below.
PHUGOID MODE
Lanchester's original solution (Lanchester, 1908) for the phugoid used the assump tions that a, = 0, A a = 0 and T  D = 0 (see Fig. 2.1). It follows that there is no net aerodynamic force tangent to the flight path, and hence no work done on the vehicle except by gravity. The motion is then one of constant total energy, as suggested previ ously. This simplification makes it possible to treat the most general case with large disturbances in speed and flightpath angle (see Miele, 1962, p. 271 et seq.) Here we content ourselves with a treatment of only the corresponding smalldisturbance case, for comparison with the exact numerical result given earlier. The energy condition is
E = $mV2  mgz, = const
v2 = u; + 2gzE
where the origin of F E is so chosen that V = uo when z, = 0. With cr constant, and in addition neglecting the effect of q on C,, then C, is constant at the value for steady horizontal flight, that is, C, = C, = C,, and L = cW0$pV2S or, in view of (6.3,1),
Thus the lift is seen to vary linearly with the height in such a manner as always to drive the vehicle back to its reference height, the "spring constant" being
The equation of motion in the vertical direction is clearly, when T  D = 0, and y =
angle of climb (see Fig. 2.1)
W Lcos y = mz,
or for small y,
On combining (6.3,2) and (6.3,4) we get
which identifies a simple harmonic motion of period
Since Cwo = mgl$pu:S, this becomes
when uo is in fps, a beautifully simple result, suggesting that the phugoid period de pends only on the speed of flight, and not at all on the airplane or the altitude! This
6.3 Approximate Equations for the Longitudinal Modes 173
elegant result is not only of historical interestit actually gives a reasonable approxi mation to the phugoid period of rigid airplanes at speeds below the onset of signifi cant compressibility effects. Thus for the B747 example, the Lanchester approxima tion gives T = 107s, a value not very far from the true 93s. It is possible to get an even better approximation, one that gives an estimate of the damping as well. Be cause q is approximately zero in this mode, we can infer that the pitching moment is approximately zerothat the airplane is in quasistatic pitch equilibrium during the motion. Moreover the pitching moment can reasonably be simplified in these circum stances by keeping only the first two terms on the right side of (4.9,17e). Because q and w are both relatively small we further neglect Z, and ZG as well. On making these simplifications to (4.9,18) and setting Af, = 0 and 8, = 0 we get the reduced system of equations:
These equations are not in the canonical form x = Ax, but we can still get the charac teristic equation by substituting x = x,,eA' and factoring out the exponential. The re sult is
Equation (6.3,7) expands to
which is a convenient way to write the characteristic equation of a secondorder sys tem. The constants are
from which we derive the radian frequency and damping to be
174 Chapter 6. Stability of Uncontrolled Motion
When Mu = 0, these reduce to:
A = uoMw
from which
When Zu from Table 4.4 with C,,, = 0 is substituted into (6.3,12) we find (see Exer cise 6.1) that T,, = 2 d w , is exactly the Lanchester period (6.3,5). Moreover if we make the further assumption that the airplane is a jet with constant thrust (aTlau, =
0 ) we find the damping ratio to be
that is, it is simply the inverse of the U D ratio of the airplane. In fact the approxima tion for the period is good over the whole range of CmU, whereas that for the damping is poor for large positive C,". For the example airplane the above approximation gives < = 0.066, compared with the exact value 0.049.
SHORTPERIOD MODE
Figure 6.3b shows that the shortperiod mode is essentially one with two degrees of freedom, the speed being substantially constant while the airplane pitches relatively rapidly. We can therefore arrive at approximate system equations by neglecting the Xforce equation entirely and putting Au = 0. Examination of the magnitudes of the terms in the numerical example shows that ZG is small compared to m and Zq is small compared to mu,. The result after simplifying (4.9,18) with 80 = 0 is a pair of equa tions for w and q.
The characteristic equation of (6.3,13) is found to be
6.4 General Theory of Static Longitudinal Stability 175
When converted with the aid of Tables 4.1 and 4.4, (6.3,14) becomes
h 2 + g h + c = o (a)
where
B =    1 :, [ ;; + i, (C". + C"J I (b) (6.3,15)
1 c =   c m CCn t*2f, (cma  A) 2~ (c)
When the data for the B747 is substituted into (6.3,15) the result obtained is
h2 + 0.741A + 0.9281 = 0
with roots h = 0.371 + 0.889
which are seen to be almost the same as those in (6.2,3) obtained from the complete matrix equation. The shortperiod approximation is actually very good for a wide range of vehicle characteristics and flight conditions.
6.4 General Theory of Static Longitudinal Stability
In Chap. 2 we used positive pitch stiffness (negative C,,) as an approximate criterion for static longitudinal stability. Now static instability really means the presence of a real positive root of the characteristic equation, and we saw in Sec. 6.1 that the condi tion for no such root to occur is that the coefficient E of the stability quartic must be positive. Thus the boundary between static stability and instability is defined by E =
0. We get E by putting A = 0 in the characteristic determinant. Thus
E = detA (6.4,l)
In evaluating (6.4,l) for the matrix of (4.9,18) we put 13, = 0, and as in Sec. 6.3, we neglect the two derivatives Z;, and Z,. The result is
g E = p (Zu Mw  MuZ,) (6.42)
mI,
Since g, rn and I , are all positive, the criterion for static stability is
Z,M,  M,Z, > 0. (6.43)
When converted to nondimensional form, this becomes
Cma(Czu  2Cw0)  Cm,,Czu > 0 (6.4,4)
When there are no speed effects, that is, Czu and CmU are both zero, then the criterion does indeed reduce to the simple CmU < 0.
We now compare the above criterion for stability with the trim slope (2.4,24). In making this comparison, we must take note of a minor difference in basic assump tions. In the preceding development, it was specifically assumed that the thrust vector rotates with the vehicle when CY is changed [see (5.1,1)]. In the development leading
176 Chapter 6. Stability of Uncontrolled Motion
to (2.4,24) by contrast, there is an implicit assumption that the thrust provides no component of force perpendicular to V [see (2.4,18)]. It is this difference that leads to the presence of C,, in (6.4,4) instead of CL, in (2.4,24). Had the assumptions been the same, the expressions would be strictly compatible. In any case, C,, is usually small compared to C,,, SO that the difference is not important, see Table 5.1. We see that the justification for the statement made in Sec. 2.4, that the slope of the elevator trim curve ( d ~ , , , ~ , l d ~ ) ~ is a criterion of static stability, is provided by (6.4,4). [Note that Cwo = C, in (6.4,4).]
Another stability criterion referred to in Chap. 2 is the derivative dCmldC, (2.3,8). It was pointed out there that this derivative can only be said to exist if enough constraints are imposed on the independent variables a , V, a,, q, etc., on which Cm and CL separately depend. Such a situation results if we postulate that the vehicle is in rectilinear motion (q = 0 ) at constant elevator angle and throttle setting, with L = W, but with varying speed and angle of attack. Such a condition cannot, of course, actually occur in flight because the pitching moment could be zero at only one speed, but it can readily be simulated in a wind tunnel where the model is restrained by a balance. With the above stipulations, Cm and CL reduce to functions of the two vari ables ic and a , and incremental changes from a reference state ( ), are given by
dCL = CLm da + CLu dii
dCm = Cm, da + Cmu dii
The required derivative is then
dii Cma + Cmu
dii CL, + CL" 
da
provided diilda exists. This is guaranteed by the remaining condition imposed, that is, L = W (implying a , = 0). For then we have
W = CL(a, 2)hpv2S = const
from which we readily derive
(CLa d a + CLu dii)ipu;S + CL,pu$ du = 0
From (6.4,7)
(CL" + 2CL,) dii + CL, da = 0
dii  =  CL, d a C L ~ + ~ C L ,
After substituting (6.4,8) into (6.4,6) and simplifying we get
On comparing (6.4,9) with (6.4,4), with the same caveats as for the trim slope, we see that the static stability criterion is
6.5 Effect of Flight Condition on the Longitudinal Modes of a Subsonic Jet Transport 177
provided that dC,ldC, is calculated with the constraints A6, = A6, = q = 0 and L = W. [The quantity on the left side of (6.4,9) is sometimes referred to as speed stability in the USA, by contrast with "angle of attack" stability. In Great Britain, this term usually has a different meaning, as in Sec. 8.5.1
On using the definition of h, given in (2.4,26) we find from (6.4,9) that
that is, that it is proportional to the "stability margin," and when CLu + 2CL,, is equal to it.
6.5 Effect of Flight Condition on the Longitudinal Modes of a Subsonic Jet Transport
In Sec. 6.2 we gave the representative characteristic modes of a subsonic jet airplane for a single set of parameters. It is of considerable interest to enquire into how these characteristics are affected by changes in the major flight variablesspeed, altitude, angle of climb, and stability margin. It is also of interest to look into the effect of the vertical density gradient in the atmosphere. In this section and by means of exercises we examine some of these effects.
EFFECT OF SPEED AND ALTITUDE
The data in Heffley and Jewel (1972) for the example airplane include several com parable cases, all having the same geometry, static margin, and gross weight. There are two speeds at sealevel, and three each at 20,000 and 40,000ft altitudes. The modal periods and damping for these eight cases are displayed in Fig. 6.5. (Since there are so few points, the shapes of the curves are conjectural!) It is an understate ment to say that there is no simple pattern to these data. It can be said, however, that the phugoid period increases with speed, as predicted by the Lanchester theory, and decreases with altitude at fixed Mach number. The shortperiod does the opposite, de creasing with speed and increasing with altitude.
The most striking feature of the data is the sudden and large increase in the phugoid period at high Mach number at the two higher altitudes. This phenomenon is a result of a loss of true static stability at these Mach numbers brought about by a negative value of CmL<, which has the effect of reducing E in (6.4,2). This happens be cause this large aircraft is necessarily quite flexible, and because at these Mach num bers it is entering the transonic regime, where air compressibility leads to substantial alterations in the aerodynamic pressure distribution. To show that CmU is the reason for the behavior of the graphs, we vary it over a large range for the flight condition M = 0.8 and 20,000ft altitude. Figure 6.6 shows the result and substantiates the impor tant role of this derivative. In fact, from (6.4,4) we calculate that E = 0 when CmU =
0.0968, a value only 4% more negative than that of the example at the given flight
Chapter
Mach number
(a) Phugoid mode
2t Period (s)
0 1 I 1 I I I 0.4 0.5 0.6 0.7 0.8 0.9 1
Mach number
(b) Shortperiod mode
Figure 6.5 Variation of longitudinal modes with speed and altitude. (a) Phugoid mode. (b) Short period mode.
6.5 Effect of Flight Condition on the Longitudinal Modes of a Subsonic Jet Transport 179
7000 1 @ B 747 example
/Exact
,Lanchester approx.
I Approx. (6.3.1 0) '
0.1 0.05 0 0.05 0.1
Cmu
(b ) Damplng
Figure 6.6 Effect of C,,, on the phugoid mode. (a ) Period. (b) Damping.
180 Chapter 6. Stability of Uncontrolled Motion
condition. Thus at this point the airplane would be very close to the static stability boundary, at which the period would go asymptotically to m.
EFFECT OF VERTICAL DENSITY GRADIENT
We might expect on physical grounds that the vertical gradient in atmospheric den sity would have an effect on the phugoid mode. For when the airplane is at the bot tom of a cycle and moving fastest it is also in air of greater density and hence would experience an additional increase in lift. It turns out that this effect is appreciable in magnitude. We shall therefore do two things: (1) show how to include this effect in the general equations of motion, and (2) derive a representative order of magnitude of the change in the phugoid period.
The modification to (4.9,18) consists of moving the Ai , equation into the matrix equation and adding some appropriate derivatives to the aerodynamic forces (4.9,17). Since the only possible steady reference state in a vertically stratified atmosphere is horizontal flight, we take 8, = 0. If A denotes the original system matrix, the result is
In (6.5,l) there are three new derivatives with respect to 2,. Consider Z, first:
It is reasonable to neglect the variation of C, with p, and the density varies exponen tially with height,2 so that
and
where K is constant over a sufficient range of altitude for a linear analysis. It follows that
'Exactly, in an isothermal atmosphere of uniform composition; approximately, in the real atmos phere.
6.5 Effect of Flight Condition on the Longitudinal Modes of a Subsonic Jet Transport 181
Similarly,
From (4.9,6) we get the reference values, leading to
The result is a rather simple elaboration of the original matrix equation. To get an es timate of the order of magnitude of the density gradient effect, it is convenient to re turn to the Lanchester approximation to the phugoid, and modify it to suit.
In Sec. 6.3 we saw that with this approximation, there is a vertical "spring stiff ness" k given by (6.3,3) that governs the period. When the density varies there is a second "stiffness" k' resulting from the fact that the increased density when the vehi cle is below its reference altitude increases the lift, and vice versa. This incremental lift associated with a density change is
so that
Using (6.5,3) we get
Thus we find that kt is approximately constant, whereas k from (6.3,3) depends on C,,p, which varies as v  ~ for constant weight. The density gradient therefore has its greatest relative effect at high speed. The correction factor for the period, which varies inversely as the square root of the stiffness, is
so that the period, when there is a density gradient, is T' = FT. With the given values of k and kt this becomes
in which the principal variable is seen to be the speed. Using a representative value for K of 4.2 X (6.5,8) gives a reduction in the phugoid period of 18% for the ex ample airplane at 774 fps. This is seen to be a very substantial effect. If the full sys
182 Chapter 6. Stability of Uncontrolled Motion
Exact
(6.3.1 0)
0 1 I I I I 10 0 0.05 0.1 0.15 0.2 0.25
Static margin, K,,
Figure 6.7 Variation of period and damping of phugoid mode with static margin.
tem model (6.5,l) is used, comparable effects can be found on the damping of the phugoid as well.
EFFECT OF CG LOCATION
It was indicated in Chap. 2 that the single most important aerodynamic characteristic for longitudinal stability is the pitch stiffness Cma, and that it varies strongly with the CG position, that is,
where the static margin is K,, = h,  h. The effect of this parameter is demonstrated by using (4.9,18) with variable K,. The results, with all the other numerical data iden tical with that in Sec. 6.2, are shown in Figs. 6.76.9. Figure 6.7 shows that the phugoid period and damping vary rapidly at low static margin and that the approxi mation (6.3,10) is useful mainly at large K,,. Figure 6.8 shows the variation of the
< Exact and (6.3.15)
Static margin, K, Figure 6.8 Variation of period and damping of shortperiod mode with static margin.
( c )
Figure 6.9 (a) Locus of shortperiod roots, varying static margin. (b) Locus of phugoid roots. varying static margin. (c) Locus of phugoid roots, varying static margin, Mu = 0.
.40
.30
.20
"7 . l o  ??
0 3
 10
.20
.30
.40
.50
Root locus  
Osc~l la t~on branch
 0
0.02 0  . , C  Subs~dence
branch 


I I I 1
0 0 04 Static margin
/ , h n  h, 0 02
I\ 0.0075 0 0.02      .  A B
0.07
0 0.02
0 0.04
1 1 I
184 Chapter 6. Stability of Uncontrolled Motion
shortperiod roots. These, too, vary strongly with pitch stiffness, the mode becoming nonoscillatory at a static margin near zero. The approximation (6.3,15) is excellent over the whole oscillatory range.
Important additional insight into these modes is obtained by examining the root loci obtained by varying the static margin. These are shown in Fig. 6.9. Figure 6 . 9 ~ shows that the damping, n, of the shortperiod mode remains virtually constant with decreasing K,, while the frequency, w, decreases to zero at point A where the locus splits into two real roots, branches AB and AC of the locus. These of course represent nonperiodic modes or subsidences. Figure 6.9b displays the much more complex be havior of the phugoid. With reducing static margin (rearward movement of the CG) this mode becomes unstable at point D. At (totally unrealistic!) negative static margin beyond 0.1, a new stable oscillation has reappeared. However, it is accompanied by a catastrophic positive real root far to the right.
The importance of the Mu derivative was shown earlier. It is again displayed in Fig. 6.9c, which repeats the locus of the phugoid roots with Mu set equal to zero. The corresponding locus for the shortperiod roots is almost identical to Fig. 6.9a. The pattern with Mu = 0 would be more representative of a rigid airplane at low Mach number. It shows a stable phugoid at all static margins as K,, is decreased until it splits at point D into a pair of real roots. The left branch from D then interacts with the branch AB of the shortperiod locus to generate a new stable oscillation while the right branch crosses the axis to give an unstable divergence at negative static margin.
6.6 Longitudinal Characteristics of a STOL Airplane
The curves of Fig. 6.5 show that the characteristic modes of an airplane vary with speed, that is with the equilibrium weight coefficient Cw,. In particular, the two char acteristic periods begin to approach one another as Cwo becomes large. It is of inter est to explore this range more fully by considering a STOL airplane, operating in the "poweredlift" region for which Cwo may be much larger. To this end the data given in Margason et al., (1966) has been used to obtain a representative set of coefficients for 2.0 5 Cwo 5 5.0. The flight condition assumed is horizontal steady flight, so that Cxo = 0. (The particular data used for the reference was that for the aircraft with a large tail in the high position, it = 0, and af = 45O.) From the given curves, and from crossplots of the coefficients C,, C,, and C, vs. C, at constant a , the data in Table 6.5 were derived for the equilibrium condition. Smooth curves were used for interpo lation. Since this is not a tiltwing airplane, a, is not large in the cases considered, and has been assumed to be zero.
Since aeroelastic and compressibility effects are negligible at the low speeds of STOL flight the required speed derivatives are given by (see Table 5.1)
For a propellerdriven airplane, the value of C, is given by (5.3,6), and an examina tion of the data for a typical constantspeed propeller at low speed3 showed that aTlau is very small. Hence we have used C, = 2CT0 in this example.
3The De Havilland Buffalo airplane.
6.6 Longitudinal Characteristics of a STOL Airplane 185
Table 6.5 Basic Data for STOL Airplane
Using the formulae of Table 5.1, the following estimates were made of the q and ix derivatives:
Finally the following inertial and geometric characteristics were assumed:
With the above data, the coefficients of the system matrix were evaluated, and its eigenvalues and eigenvectors calculated. The main results are shown on Figs. 6.106.13. Figures 6.10 and 6.1 1 show the loci of the roots as C,, varies between 2 and 5. The effect of C,,, is seen to be large on both modes, the shortperiod mode be coming nonoscillatory at a value of C,,, somewhat greater than 3.5, and the damping
100 wt*
B
100 nt* 1: 1 5
Figure 6.10 Root locusshortperiod mode, STOL airplane.
 0.5 I I I I 1 I I
0 > 100 nt* 0.7 0.6 0.5 0.4 0.3 0.2 0.1  0.5
Cwo  1.0
 1.5  2.0
 2.5
Figure 6.11 Root locusphugoid mode, STOL airplane.
I
of the phugoid increasing rapidly at the same time. Figure 6.12 shows the two peri ods, and that they actually cross over at C,,, = 3.4. The concept of the phugoid as a "long" period oscillation is evidently not applicable in this situation! The approxima
I tions to the phugoid and the pitching mode are also shown for comparison. It is seen that they give the two periods quite well, and that (6.3,15) also depicts quite accu rately the damping of the pitching oscillation and of the two nonperiodic modes into which it degenerates at high C,,. The phugoid damping, however, is not at all well predicted by the approximate solution. Figure 6.13 shows that the modes are all heav ily damped over the whole range of C,.
o ! I I I J
2 3 4 5
Cwo
Figure 6.12 Periods of oscillatory modes, STOL airplane.
186 Chapter 6. Stability of Uncontrolled Motion
6.7 Lateral Modes of a Jet Transport 187
I I I I I
   . Nonoscillatory modes ( x 10)
o l I I I I 1 .O 2.0 3.0 4.0 5.0
Figure 6.13 Time to damp of modes, STOL airplane.
6.7 Lateral Modes of a Jet Transport
We use the same airplane and flight condition as for the longitudinal modes in Sec. 6.2, and calculate the lateral modes. The nondimensional and dimensional derivatives are given in Tables 6.6 and 6.7. Using these, the system matrix of (4.9,19) is found to be (note that the state vector is [v p r + I T . ) :
This yields the characteristic equation
The stability criteria are
E = 0.003682 > 0
R = 0.04223 > 0
so there are no unstable modes.
Table 6.6 Nondimensional DerivativesB747 Airplane
188 Chapter 6. Stability of Uncontrolled Motion
Table 6.7 Dimensional DerivativesB747 Airplane
EIGENVALUES
The roots of (6.7,2) are
Mode 1 (Spiral mode): A, = 0.0072973
Mode 2 (Rolling convergence): A, = 0.56248
Mode 3 (Lateral oscillation or Dutch Roll): A , , = 0.03301 1 2 0.946551'
Table 6.8 shows the characteristic times of these modes. We see that two of them are convergences, one very rapid, one very slow, and that one is a lightly damped oscilla tion with a period similar to that of the longitudinal shortperiod mode.
EIGENVECTORS
The eigenvectors corresponding to the above eigenvalues are given in Table 6.9. In addition to the basic 4 state variables, Table 6.9 contains two extra rows that show the values of the two state variables @ and y, (see Exercise 6.2).
MODE 1: THE SPIRAL MODE
From Table 6.9, we find the ratios of the angle variables in the spiral mode to be
P:+:$ = 0.00119:0.177:l
so that the motion is seen to consist mainly of yawing at nearly zero sideslip with some rolling. This is, of course, the condition for a truly banked turn, and this mode
Table 6.8 Characteristic TimesLateral Modes
Mode Name Period (s) thav(s) Nhotf
1 Spiral  95  2 Rolling  1.23
convergence 3 Lateral oscillation 6.64 21 3.16
(Dutch Roll)
6.7 Lateral Modes of a Jet Transport 189
Table 6.9 Eigenvectors (polar form)
Spiral Rolling convergence Dutch Roll
Magnitude Phase Magnitude Phase Magnitude Phase
can be thought of as a variableradius turn. The aerodynamically important variables are
P:@:i = 1:0.137:0.773
and the largest of these, P, has already been seen to be negligibly small for moderate values of 4 and $. The aerodynamic forces in this mode are therefore very small, and it may be termed a "weak" mode. This is consistent with its long time constant.
The flight path in the spiral mode can readily be constructed for any given initial yaw angle from the eigenvector. For example, with an initial $ of 20' (0.35 rad), we have from Table 6.9
$ = 0.35eA1'
u = 774(0.001 19)0.35eA1' fps
where A , = 0.0072973 s'
From (4.9,19) and the above it follows that (for 0,) = 0)
y, =  37079~0.007*~73' ft
x, = 774 t ft
Figure 6.14 shows the pathit is seen to be a long, smooth return to the reference flight path, corresponding to y, = 0. When the spiral mode is unstable as is fre quently the case, $, 4, and y, are all of the same sign, and of course all increase with time instead of decreasing, as shown in the figure.
MODE 2: THE ROLLING CONVERGENCE
The ratios of the angle variables in this mode are, from Table 6.9,
The mode is evidently one of almost pure rotation around the x axis, and hence its name. The variables that are significant for aerodynamic forces are (P, 9, i ) and they are in the ratios
P:@i = 0.278: 1:0.0561
190 Chapter 6. Stability of Uncontrolled Motion
Figure 6.14 Flight path in spiral mode.
so that the largest rolling moment in this mode of motion is CIJ, and the i contribu tions are negligible by comparison.
MODE 3: THE LATERAL OSCILLATION (DUTCH ROLL)
The vector diagram for this mode is shown in Fig. 6.15. It is seen that the three angle variables P, 4, + are of the same order of magnitude, that i is an order smaller, and
; = 0.037 (not vis~ble)
Figure 6.15 Vector diagram of lateral oscillation. C,, = 0.57. Altitude = 40,000 ft.
6.7 Lateral Modes o f a Jet Transport 191
that p and cC, are almost equal and opposite. It follows from (4.9,19) that j , is nearly zero. In dimensional terms, when 14) = 20°, JyEJ = 8 ft, whereas the wavelength of the oscillation is about 5000 ft. The vehicle mass center is seen to follow a nearly rectilinear path in this mode, the motion consisting mainly of yawing and rolling, the latter lagging the former by about 160" in phase.
EFFECT OF SPEED AND ALTITUDE
Even for the "basic" case of a rigid airplane at low Mach number, the variation of the lateral modes with speed and altitude may not be simple. This is because some of the lateral stability derivatives are dependent on the lift coefficient in complex ways. That is especially true of airplanes with swept wings and low aspect ratio for which C,, in creases markedly with C,. These effects will appear most strongly at low speed and high altitude, both of which require high C,. (Note that in the B747 example at M =
0.8 and 40,000 ft, C, = 0.654, which is quite large for cruising flight.) For a rigid sweptwing airplane at low Mach number the period of the Dutch Roll mode would be expected first to increase and then to decrease as the airplane speed increases. The damping of this mode would be expected to be weak at low speed and to increase at higher speeds. The rolling convergence is well damped at all speeds, but the damping would normally increase with speed. The spiral mode is frequently unstable over some portion of the speedlaltitude flight envelope, depending on the interplay of the derivatives that appear in (6.8,6). The characteristic times of this mode are, however, usually so long that the instability does not degrade the handling qualities unduly.
The effect of increasing altitude at fixed C, is primarily an increase in the damp ing time constants of all the modes. The period of the Dutch Roll is not much af fected.
When substantial aeroelastic and compressibility effects are added to the already complex behavior of the lateral modes, the result is an even more irregular pattern of modal characteristics. The data of (Heffley and Jewel, 1972) for the B747, repro duced in Table 6.10 show this. (Note that a negative t,,,,, implies an unstable mode.) At the two lower altitudes, with relatively low values of C,, the modes are seen to be have in a fairly regular way (see Fig. 6.16). However, at 40,000 ft and high Mach
Table 6.10 Variation of Lateral Modes with Speed and Altitude
Spiral Rolling Lateral oscillation mode convergence (Dutch Roll)
Altitude, Mach thdj th& Period Nttalf
ft No. ( 5 ) ( 3 ) (3) (cycles)
192 Chapter 6. Stability of Uncontrolled Motion
Mach number
(a) Lateral oscillation
0 1 1 I I I I I 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Mach number
(b ) Rolling convergence
Figure 6.16 Variation of lateral modes with speed and altitude. (a ) Lateral oscillation. (b) Rolling convergence. (c ) Spiral mode.
6.8 Approximate Equations for the Lateral Modes 193
Altitude 20,000 ft
Mach number
(c) Spiral mode
Figure 6.26 (Continued)
number the lateral behavior is quite irregular, especially the variation of the damping of all the modes with M. The spiral mode is seen to be unstable (albeit with long time constant) at both M = 0.7 and 0.9 but stable at M = 0.8. This behavior is primarily the result of the complex variation of C,, with C, and M in this region.
6.8 Approximate Equations for the Lateral Modes
As with the longitudinal modes we should like if possible to have useful analytical approximations to the lateral characteristics. We find that there are reasonable ap proximations to all three modes, but the application of all such approximations must be made with caution. Their accuracy can really be verified only a posteriori, by comparison with exact solutions. They can only be used with confidence in situations similar to those in which they have previously been found to work well.
SPIRAL MODE
Comparison of the eigenvalues in Sec. 6.7 shows that A for the spiral mode is two or ders of magnitude smaller than the next larger one. This suggests that a good approx imation to this root may be obtained by keeping only the two lowestorder terms in the characteristic equation, that is,
where A, denotes the real root for the spiral mode. Before deriving expressions for D and E, we rewrite the matrix of (4.9,19) in a more compact notation for convenience, including the approximation Y,, = 0.
194 Chapter 6. Stability of Uncontrolled Motion
3, 0 3,. g cos 8, 2, zp 2 r
Nu NP Nr o 0 1 tan 8, 0 o 1
The meanings of the symbols in (6.8,2) are obtained by comparison with (4.9,19), for example
L, 2, = , + 1 2 ,
I,
and in the special case when the stability axes are also principal axes, I, = 0 and
With the notation of (6.8,2), expanding det (A  AI) yields
E = g[(2,Nr  .%,Nu) cos 8, + (YpN,  2,Np) sin 00] (a)
D = g(2, cos 8, + N, sin 8,) + 9.(2,.%  TpNr) (6.8,3)
+ 9r(2PJfu  2,NP> (b)
When the orders of the various terms in D are compared, it is found that the second term can be neglected entirely and Y, can be neglected in 9,. The approximation that then results is
D = g(2, cos 8, + Nu sin 8,) + uo(Y,Np  YPNu) (6.89)
The result obtained from (6.8,1), (6.8,3a), and (6.8,4) for the jet transport example of Sec. 6.7 is A, = 0.00725, less than 1% different from the correct value. Equation (6.8,l) is seen to give a good approximation in this case.
It will be recalled that the coefficient E has special significance with respect to static stability (see Sec. 6.1). We note here that in consequence of (6.8,l) the spiral mode may exhibit exponential growth, and that the criterion for static lateral stability is
(TUNr  2,N.) cos 8, + (2,N,  Y,Np) sin $ > 0 (6.8,5)
On substituting the expanded expressions for 2, and so forth, (6.85) reduces to
(ClpCn,.  C1,Cnp) cos 80 + (C~~cnp  C~pcn,,) sin '30 > 0 (6.8,6)
Since some of the derivatives in (6.8,6) depend on C,,, the static stability will vary with flight speed. It is not at all unusual for the spiral mode to be unstable over some portion of the flight envelope (see Table 6.10 and Exercise 6.3).
ROLLING M O D E
It was observed in Sec. 6.7 that the rolling convergence is a motion of almost a single degree of freedom, rotation about the xaxis. This suggests that it can be approxi mated with the equation obtained from (4.9,19) by putting v = r = 0, and consider ing only the second row, that is,
6.8 Approximate Equations for the Lateral Modes 195
which gives the approximate eigenvalue
The result obtained from (6.8,8) for the B747 example is A, = 0.434, 23% smaller than the true value 0.562. This approximation is quite rough.
An alternative approximation has been given by McRuer et al. (1973). This ap proximation leads to a secondorder system, the two roots of which are approxima tions to the roll and spiral modes. In some cases the roots may be complex, corre sponding to a "lateral phugoidma longperiod lateral oscillation. The approximation corresponds to the physical assumption that the sideforce due to gravity produces the same yaw rate r that would exist with /3 = 0. Additionally Y,, and Yr are ne glected. With no approximation to the rolling and yawing moment equations the sys tem that results for horizontal flight is
The procedure used to get the characteristic equation of (6.8,9) is the same as that used previously for the phugoid approximation in Sec. 6.3. The result is
which expands to
where
The result of applying (6.8,l 1 ) to the B747 example is
A, = 0.00734 and A, = 0.597
These are within about 1% and 6% of the true values, respectively, so this is seen to be a good approximation for both modes, certainly much better than (6.8,7) for the rolling mode.
DUTCH ROLL MODE
A physical model that gives an approximation to the lateral oscillation is a "flat" yawinglsideslipping motion in which rolling is suppressed. The corresponding equa tions are obtained from (4.9,19) by setting p = 4 = 0 and dropping the second (rolling moment) equation. The term in Y,. is also neglected in the first equation. The
196 Chapter 6. Stability of Uncontrolled Motion
result is
The corresponding characteristic equation is readily found to be
.The result obtained from (6.8,12) for our example is A,, = 0.1008 f 0.9157i, or
T = 6.86 sec
N,,,,, = 1 .o
The approximation for the period is seen to be useful (an error of about 3%) but the damping is very much overestimated.
There is another approximation available for the damping in this mode that may give a better answer. It follows from the fact that the coefficient of the nexttohighest power of A in the characteristic equation is the "sum of the dampings" (see Exercise 6.4). Thus it follows from the complete system matrix of (6.8,2) that
But the approximation (6.8,ll) for the roll and spiral modes gives precisely
On using (6.8,11) we get the expression
which is to be compared with a(%, + Nr) given by (6.8,12). The damping obtained from (6.8,14) is nDR = 0.0159, better than that obtained from (6.8,12) but still quite far from the true value of 0.0330. The simple average of the two preceeding ap proximations for the Dutch Roll damping has also been used. In this instance it gives n, = 0.0584, which although better is still 77% off the true value.
This example of an attempt to get an approximation to the Dutch Roll damping illustrates the difficulty of doing so. Although the approximation tends to be better at low values of C,, nevertheless it is clear that it must be used with caution, and that only the full system matrix can be relied on to give the correct answer.
6.9 Effects of Wind
In all the preceding examples, the atmosphere has been assumed to be at rest or to have a velocity uniform in space and constant in time. Since this is the exceptional rather than the usual case, it is necessary to examine the effects of nonuniform and unsteady motion of the atmosphere on the behavior of flight vehicles. The principal effects are those associated with atmospheric turbulence, and these are treated at some length in Etkin (1972, 1981). However, quite apart from turbulence, the wind
6.9 Effects of Wind 197
may have a mean structure which is not uniform in space, that is, there can be spatial gradients in the timeaveraged velocity. The examples of most concern are down bursts and the boundary layer next to the ground produced by the wind blowing over it. Downbursts are vertical outflows from low level clouds that impinge on the ground, somewhat in the manner of a circular jet, and spread horizontally. The result ing wind field has strong gradients, both horizontal and vertical. A number of air plane accidents have been attributed to this phenomenon.
In order to introduce wind into the analytical model, we must make any alter ations that may be needed, because of the presence of the wind, to the aerodynamic forces and moments. In the trivial case when the wind is uniform and steady, no change is necessary to the representation of aerodynamic forces from that used be fore. However, turbulence and wind gradients may require such changes.
Since the linear model that was developed in Chap. 4 is based on small distur bances from a steady reference condition, and since there is no such steady state when the aircraft is landing or taking off through a boundary layer or downburst, the linear model is of limited use in these situations. For this kind of analysis, one must use the nonlinear equations (4.7,l)(4.7,5) and introduce a model for the aerody namic forces that embraces the whole range of speeds and attitudes that will occur throughout the transient (Etkin, B. and Etkin, D. A. (1990)). Such an analysis is be yond the scope of this volume.
There is one relevant steady state, however, that can be investigated with the lin ear model, and that is horizontal flight in the boundary layer. The planetary boundary layer has characteristics quite similar to the classical flatplate turbulent boundary layer of aerodynamics. The vertical extent of this layer in strong winds depends mainly on the roughness of the underlying terrain, but is usually many hundreds of feet. Figure 6.17 shows the powerlaw profiles associated with different roughnesses. These are all of the form
w = khn (6.9,l)
where, as indicated in the figure, h is height above the ground. The vertical gradient is then given by
For example, for smooth terrain (n = 0.16), and for a wind of 50 fps at 50 ft altitude, the gradient would be dW/dh = 0.16 fpslft.
By way of example, we shall analyze what effect the vertical wind gradient has on the longitudinal modes of the STOL airplane of Sec. 6.6 in low speed flight, when wind effects can be expected to be largest. In order to generate the analytical model, we go back to the exact equations (4.7,l) et seq. The longitudinal equations, when linearized for small perturbations around a reference state of horizontal flight at speed u, and 8, = 0 are
6.9 Effects of Wind 199
The wind is specified to vary linearly with altitude with gradient r = dW/dzE and to be parallel to the xz plane. The wind vector is prescribed in frame FE as
We will assume that this implies a tailwind of strength Wo at the reference height. For a wind that increases with altitude, r = sgn(~,)lI'I. To convert WE to body axes, we use the transformation matrix L,, [the transpose of (4.4,3)] for small angles and 4 = I) = 0 to get (see Exercise 6.6)
In the third component of (6.9,5) we have neglected the secondorder product z,8. We can now use (6.9,5) together with V E = V + W to eliminate uE and wE from (6.9,3) with the result
AX  mg8 = m(Au + Ti,) AZ = m(w + +,6  utq)
We note that U: = u0 + W, and eliminate 6 and 2, from the right side of (6.9,6) with the result
AX  mg0 = m(Au + r[u,8 + w])
It is observed that the only explicit effect of wind on the system equations in this case is the term containing r on the right side of the first equation. Since a uniform wind (i.e., r = 0) does not affect the airplane dynamics the system must be indepen dent of Wo, and we see that W, does indeed not appear anywhere in the equations. In addition to what we see in (6.9,7), however, there are some implicit effects of the wind gradient on the aerodynamic derivatives.
It is clear that the changes in pressure distribution over the surfaces of a vehicle, and hence its basic aerodynamic derivatives, are not the same when the incident flow has a gradient I' as when it is spatially uniform. Two simple examples suffice to make this clear. (1) When there is a perturbation A a from the reference state, the tail moves downward into a region of lower air velocity and on this account one would expect ~~~~1 to be smaller than normal. (2) When the wing rolls through an angle 4, the right tip moves into a lowwind region, and the left tip into a highwind region. The gradi ent in velocity across the span is like that associated with yaw rate r, and hence we should expect values of C,, and C,,, proportional to the wing contributions to Clr and
200 Chapter 6. Stability of Uncontrolled Motion
Cnr. Note that for upwind flight this leads to an unstable roll "stiffness" C,, > 0 where none existed before (C,, is negative for downwind flight).
Reasonable estimates of the major changes in the basic derivatives associated with r can be made from available aerodynamic theories, but a complete account of these is not currently available, and to develop them here would take us too far afield. Instead we simply incorporate the additional terms given by (6.9,7) into the longitu dinal equations of motion and note the extent of the changes they make in the charac teristic modes previously calculated. The appropriate matrix for this case is obtained from (4.9,18) by adding the two terms containing T, and is given by (6.9,s).
In (6.9,8), A' is the 3 X 4 matrix consisting of the last three rows of the matrix of (4.9,18), with 8, = 0.
An example of the results for the STOL airplane is shown in Figs. 6.18 and 6.19. The numerical data used was the same as in Sec. 6.6, with wind gradient variable from 0.30 fpslft (the headwind case) to +0.30 fpslft (the tailwind case). The effects on both the phugoid and pitching modes are seen to be large. A strong headwind de creases both the frequency and damping of the phugoid, and a strong tailwind changes the real pair of pitching roots into a complex pair representing a pitching 0s cillation of long period and heavy damping.
100 wt'
4
I I I I I 0
* 100 nt* 0.5 0.4 0.3 0.2 0.1
Figure 6.18 Effect of wind gradient on phugoid rootsSTOL airplane. C, = 4.0.
6.10 Exercises 201
100 wt*
'Headwind I I
Figure 6.19 Effect of wind gradient on shortperiod rootsSTOL airplane. C , = 4.0.
6.10 Exercises
6.1 Use (6.3,10) to calculate the approximate period T, = 2 d o n of the phugoid oscilla tion. Assume Mu = 0.
6.2 Derive the entries for cC, and (y,lu,t *) in Table 6.9 from those for 0 and P
6.3 The stability derivatives of a general aviation airplane are given in Table 7.2. The air plane weighs 2400 lb (10,675 N) and has a wing area of 160 ft2 (14.9 m2). The flight altitude is sea level. Calculate and plot the spiral stability criterion E as a function of speed (0.15 < C, < 1.7) for values of 8, =  lo0, 0°, 10".
6.4 The characteristic equation is of order N. Prove that the coefficient of AN' is the neg ative of the sum of the real parts of all the roots, and hence is aptly termed "the sum of the dampings."
6.5 Find the critical climb angle for spiral stability of the jet transport of Sec. 6.7. [Hint: start with (6.8,6)]. Having regard to its expected influence on the stability derivatives, state the effect on spiral stability in horizontal flight of increasing the wing dihedral angle.
6.6 Carry out the transformation W, = LBEWE to get (6.9,5).
6.7 Using the stability derivatives given in Table 7.2 for a general aviation airplane, cal culate the lateral modes in the absence of gravity. The relevant data are:
W = 2400 lb (10,675 N) I, = 170 slugft2 (230 kg.m2)
S = 160 ft2 (14.9 m2) I, = 1,312 slug.ft2 (1,778 kg.m2)
b = 30 ft (9.14 m) I, = 0
202 Chapter 6. Stability of Uncontrolled Motion
V = 150 knots (77.3 mls) 80 = 0
altitude = sea level
Compare the results with those for gravity present.
6.8 Find the characteristic equation of the hovercraft of Exercise 4.10. Show that when it is statically unstable with both M , and L, positive it can be gyrostabilized (like a spinning top, i.e., solutions remain bounded) if H is large enough.
6.9 A conventional stable aircraft is on a steady descent to a landing on a shallow glide slope when the headwind suddenly vanishes. What initial condition problem de scribes the subsequent motion? Describe qualitatively, from your knowledge of longi tudinal natural modes, what the subsequent flight path will be if the elevator and throttle controls remain fixed at their prior positions.
6.10 Show that if we neglect all the Y force derivatives, then the smalldisturbance equa tions (for 8, = 0) yield the following approximation for the lateral displacement:
6.11 Assume that in the lateral oscillation of an airplane, the modal diagram shows that 4 > 14 and the t,!i vector leads the 4 vector by an angle between 90" and 180". This is reasonably representative of flight at low Mach number. Describe the relative mo tion of the two wing tips. To simplify the situation, neglect damping, the motion asso ciated with sideslip and with the forward motion of the airplane, assume the angles are "small," and interpret "wing tip" to mean a point on the yaxis.
Hold a model airplane by its wing tips and practise until you can execute a motion of the type you have deduced. Observe it in a mirror. (The name Dutch Roll was given to this motion because of a perceived resemblance to that of an ice skater, Holland being noted for this sport.)
6.12 The theory for a stable airplane shows that if the controls are neutral it will fly in a steady state on a straight line at constant speed. However, the development in the text assumed that the airplane produced no lateral aerodynamic force or moments when the controls were set to neutral and the lateral state variables were zero. Real air planes cannot achieve this owing to design and construction factors. You are asked in this exercise to examine the steady states that are possible for such an airplane. The steady state here is defined to be one in which the linear and angular velocity components (u, v , w, p, q, r), the aerodynamic forces and moments (X, Y, Z, L, M, N ) and the gravity force components (X,, Y,, 2,) are all constant.
(a) Show that in the steady state the two Euler angles 8 and 4 are also constant. (b) Since 8 is constant, it can be set to zero by a suitable choice of body axes. For
this case show that p = 0 and the angular velocity vector m is vertical in F,.
(c) Starting with (4.5,8) and (4.5,9) develop the lateral steady state equations. As sume that in the steady state q, r, 4 and v are small quantities. Assume that the controls are neutral and thus (since p = 0 in the steady state)
6.11 Additional Symbols Introduced in Chapter 6 203
where ( )a stands for effects due to lateral asymmetries in the aircraft. What condition must be met in order for a unique steady state to exist? How is this con dition related to E of (6.8,3a) when the body axes are principal axes?
(d) Relate the results of (c) to the flight of a statically stable handlaunched glider.
6.11 Additional Symbols Introduced in Chapter 6
altitude
Lull; + I ; p u
LIJI: + IIfl,, LJI: + I ; N , real part of eigenvalue
NulI; + I;.Ju
NPII; + I;$,,
NJI; + IIJ,
period of oscillation
Y,lm
Y,lm  u,
vertical wind gradient, d Wldh
density gradient parameter (see 6.5,3) eigenvalue
eigenvalue of lateral oscillation
eigenvalue of rolling convergence
eigenvalue of spiral mode
damping ratio
imaginary part of eigenvalue, circular frequency
undamped circular frequency
C H A P T E R 7
Response to Actuation of the Controls Open Loop
7.1 General Remarks
In this chapter we study how an airplane responds to actuation of the primary con trolselevator, aileronlspoiler, rudder, and throttle. These are of course not the only controls that can be incorporated in the design of an airplane. Also used, less fre quently, are vectored thrust and direct lift control. Closely related to the controlre sponse problem is the response of the airplane to an inflight change of configuration such as flap deflection, lowering the undercarriage, releasing stores or armaments, deploying dive brakes, or changing wing sweep. The analysis of the response of the airplane to any of these uses methods generally similar to those that are described in the following. In the remainder of this chapter, it is assumed that there is no wind.
LONGITUDINAL CONTROL
The two principal quantities that need to be controlled in symmetric flight are the speed and the flightpath angle, that is to say, the vehicle's velocity vector. To achieve this obviously entails the ability to apply control forces both parallel and perpendicu lar to the flight path. The former is provided by thrust or drag control, and the latter by lift control via elevator deflection or wing flaps. It is evident from simple physical reasoning (or from the equations of motion) that the main initial response to opening the throttle (increasing the thrust) is a forward acceleration, i.e. control of speed. The main initial response to elevator deflection is a rotation in pitch, with subsequent change in angle of attack and lift, and hence a rate of change of flightpath direction. When the transients that follow such control actions have ultimately died away, the new steady state that results can be found in the conventional way used in perfor mance analysis. Figure 7.1 shows the basic relations. The steady speed Vat which the airplane flies is governed by the lift coefficient, which is in turn fixed by the elevator anglesee Fig. 2.19. Hence a constant 8, implies a fixed V. The flightpath angle y = 13  a, at any given speed is determined, as shown in Fig. 7.1, by the thrust. Thus the ultimate result of moving the throttle at fixed elevator angle (when the thrust line passes through the CG) is a change in y without change in speed. But we saw above that the initial response to throttle is a change in speedhence the shortterm and longterm effects of this control are quite contrary. Likewise we saw that the main initial effect of moving the elevator is to rotate the vehicle and influence y, whereas the ultimate effect at fixed throttle is to change both speed and y. The short term and longterm effects of elevator motion are therefore also quite different. The
7.1 General Remarks 205
Figure 7.1 Basic performance graph.
total picture of longitudinal control is clearly far from simple, and the transients that connect the initial and final responses require investigation. We shall see in the fol lowing that these are dominated by the longperiod, lightly damped phugoid oscilla tion, and that the final steady state with step inputs is reached only after a long time. These matters are explored more fully in the following sections.
LATERAL CONTROL
The lateral controls (the aileron and rudder) on a conventional airplane have three principal functions.
1. To provide trim in the presence of asymmetric thrust associated with power plant failure.
2. To provide corrections for unwanted motions associated with atmospheric tur bulence or other random events.
3. To provide for turning maneuversthat is, rotation of the velocity vector in a horizontal plane.
The first two of these purposes are served by having the controls generate aero dynamic moments about the x and z axesrolling and yawing moments. For the third a force must be provided that has a component normal to V and in the horizontal plane. This is, of course, the component L sin 4 of the lift when the airplane is banked at angle 4. Thus the lateral controls (principally the aileron) produce turns as a secondary result of controlling 4.
Ordinarily, the longterm responses to deflection of the aileron and rudder are very complicated, with all the lateral degrees of freedom being excited by each. Solu tion of the complete nonlinear equations of motion is the only way to appreciate these fully. Certain useful approximations of lower order are however available.
206 Chapter 7. Response to Actuation of the ControlsOpen Loop
THE CONTROL EQUATIONS
Whereas the study of stability that was the subject of Chap. 6 is generally sufficiently well served by the linear model of small disturbances from a condition of steady flight, the response of an airplane to control action or configuration change can in volve very large changes in some important variables, especially bank angle, pitch angle, load factor, speed, and roll rate. Consequently nonlinear effects may be present in any of the gravity, inertia, and aerodynamic terms. An accurate system model ca pable of dealing with these large responses must therefore begin with the more exact equations (4.7,l)(4.7,4). These would normally be reorganized into firstorder state space form for subsequent integration by a RungeKutta or other integration scheme. Equations (4.7,3df) and (4.7,4) are already in the required form. However (4.7,l) and (4.7,2) need to be rearranged. In particular, (4.7,2a and c) need to be solved si multaneously for p and i.. The functional form that results is as follows:
z c wE = f (uE, vE, p, q, 0, 4, Z') +  m
Nc i = f (p, q, r, L', N') + 7 + Ig, (c)
I , 6 = f (4, r, 4) (a)
(b) (7.193)
4 = f(p, q, r, 0, 4)
In the preceding equations, the subscript c denotes the control forces and moments, and the prime on force and moment symbols denotes the remainder of the aerody namic forces and moments. The solution of these equations would require that an aerodynamic submodel be constructed for each case to calculate the forces and mo ments at each computing step from a knowledge of the state vector, the control vec tor, the current configuration, and the wind field. To follow this course in extenso would take us beyond the scope of this text, so for the most part the treatments that follow are restricted to the responses of linear invariant systems, that is, ones de scribed by (4.9,20) with A and B constant viz
Although we are thereby restricted to relatively small departures from the steady state, these responses are nevertheless extremely useful and informative. Not only do they reveal important dynamic features, but when used in the design and analysis of automatic flight control systems that are designed to maintain small disturbances they are in fact quite appropriate.
In the examples that follow, we use {a,, 6,) for the longitudinal controls, elevator and throttle; and {So, 6,) for the lateral controls, aileron, and rudder. The aerody namic forces and moments are expressed just like the stability derivatives in terms of sets of nondimensional and dimensional derivatives. The nondimensional set is the partial derivatives of the six force and moment coefficients (C,, C,, C,, C,, C,, C,) with respect to the above control variables, such as C , , = aC,la6, or C,, = aC,laS,, and so on. The dimensional derivatives are displayed in Table 7.1.
The powerful and welldeveloped methods of modem control theory are directly applicable to this restricted class of airplane control responses. Before proceeding to specific applications, however, we first present a review of some of the highlights of the general theory. Readers who are well versed in this material may skip directly to Sec. 7.6.
7.2 Response of Linear/Znvariant Systems
For linearlinvariant systems there are four basic singleinput, singleresponse cases, illustrated in Fig. 7.2. They are characterized by the inputs, which are, respectively:
1. a unit impulse at t = 0 2. a unit step at t = 0
3. a sinusoid of unit amplitude and frequency f 4. white noise
In the first two the system is specified to be quiescent for t < 0 and to be subjected to a control or disturbance input at t = 0. In the last two cases the input is presumed to have been present for a very long time. In these two the system is assumed to be sta ble, so that any initial transients have died out. Thus in case 3 the response is also a steady sinusoid and in case 4 it is a statistically steady state. We discuss the first three of these cases in the following, but the fourth, involving the theory of random processes, is outside the scope of this text. The interested reader will find a full ac count of that topic in Etkin, 1972.
7.2 Response of Linear/Znvariant Systems 207
Table 7.1 Dimensional Control Derivatives
M
c,,,,hpu ;SF
Cm,p4pu;sZ:
Z
C z s e h ~ u 3
c,,>pu;s
8,
8,
X
C,,)PU;S
c,,>pu;s
N
C,,,,$pu;Sb
C,,,ipu;Sb
L
Cl,jpu;Sb
C,,)pu;Sb
8 ,
8,
Y
CY6 jpu t3
C,,)puiS
208 Chapter 7. Response to Actuation of the ControlsOpen Loop
(4) Figure 7.2 The four basic response problems. (1) Impulse response. (2) Step response. (3) Frequency response. (4) Response to white noise.
TRANSFER FUNCTIONS
A central and indispensable concept for response analysis is the transfer function that relates a particular input to a particular response. The transfer function, almost uni versally denoted G(s), is the ratio of the Laplace transform of the response to that of the input for the special case when the system is quiescent for t < 0. A system with n state variables xi and m controls cj would therefore have a matrix of nm transfer func tions G,,(s).
The Laplace transform of (7.1,4) is
hence
and
where
is the matrix of transfer functions. The response of the ith state variable is then given by
q(s) = 1 G,(s)Cj(s) .i
(7.233)
For a singleinput singleresponse system with transfer function G(s) we have simply
7.2 Response of Linear/lnvariunt Systems 209
..
Figure 7.3 Systems in series.
SYSTEMS IN SERIES
When two systems are in series, so that the response of the first is the input to the second, as in Fig. 7.3, the overall transfer function is seen to be the product of the two. That is,
and
Thus the overall transfer function is
Similarly for n systems in series the overall transfer function is
HIGHORDER SYSTEMS
Highorder linearlinvariant systems, such as those that occur in aerospace practise, can always be represented by a chain of subsystems like (7.2,6). This is important, because the elemental building blocks that make up the chain are each of a simple kindeither firstorder or secondorder. To prove this we note from the definition of an inverse matrix (Appendix A. 1) that
adj (sI  A) (sI  A)' =
det (sI  A)
We saw in Sec. 6.1 that det (A  sI) is the characteristic polynomial of the system. We also have, from the definition of the adjoint matrix as the transpose of the matrix of cofactors, that each element of the numerator of the right side of (7.2,7) is also a polynomial in s. (See Exercise 7.1.) Thus it follows from (7.2,2b), on noting that B is a matrix of constants, that each element of G is a ratio of two polynomials, which can be written as
in which f (s) is the characteristic polynomial. It is seen that all the transfer functions of the system have the same denominator and differ from one another only in the dif ferent numerators. Since f (s) has the roots A , . . . A,, the denominator can be factored to give
210 Chapter 7. Response to Actuation of the ControlsOpen Loop
+plrnm1tj=t+ 8 2 + a m + l s + b m + l X,
I I I I m f~rstorder 112 (n  m) secondorder components components
Figure 7.4 Highorder systems as a "chain."
Now some of the eigenvalues A, are real, but others occur in complex pairs, so to ob tain a product of factors containing only real numbers we rewrite the denominator thus
m 1 /2(n +m)
f ( s ) = n ( s  ~ , ) H ( s 2 + a p  t b r ) (7.2,lO) r= l r=m+ l
Here A, are the m real roots of f(s) and the quadratic factors with real coefficients a, and b, produce the (n  m) complex roots. It is then clearly evident that the transfer function (7.2,9) is also the overall transfer function of the fictitious system made up of the series of elements shown in Fig. 7.4. The leading component N,](s) is of course particular to the system, but all the remaining ones are of one or other of two simple kinds. These two, firstorder components and secondorder components, may there fore be regarded as the basic building blocks of linearlinvariant systems. It is for this reason that it is important to understand their characteristics wellthe properties of all higherorder systems can be inferred directly from those of these two basic ele ments.
   \, . ' .
7.3 Impulse Response
The system is specified to be initially quiescent and at time zero is subjected to a sin gle impulsive input
cj(t) = s(t) (7.391)
The Laplace transform of the ith component of the output is then
zi(s) = G,(s)B(s) which, from Table (A. l ) , item 1, becomes
xi(s) = GJs)
This response to the unit impulse is called the impulse response or impulsive admit tance and is denoted hij(t). It follows that
hU(s) = Gij(s) (a)
that is, G(s) is the Laplace transform of h(t)
CO
GJs) = Q h,(t)e' dt (6) (7.3,2)
From the inversion theorem, (A.2,11) hU(t) is then given by
7.3 Impulse Response 211
Now if the system is stable, all the eigenvalues, which are the poles of G,,(s) lie in the left half of the s plane, and this is the usual case of interest. The line integral of (7.3,3) can then be taken on the imaginary axis, s = iw, so that (7.3,3) leads to
that is, it is the inverse Fourier transform of Gij(iw). The significance of G,(iw) will be seen later.
For a firstorder component of Fig. 7.4 with eigenvalue A the differential equa tion is
x  A x = c ( 7 . 3 3
for which we easily get
The inverse is found directly from item 8 of Table A. 1 as
h(t) = eAt
For convenience in interpretation, A is frequently written as A =  1/T, where T is termed the time constant of the system. Then
A graph of h(t) is presented in Fig. 7.5a, and shows clearly the significance of the time constant T.
For a secondorder component of Fig. 7.4 the differential equation is
jj + 2604 + = c (7.33)
where x = Ly yIT is the state vector. It easily follows that
Let the eigenvalues be A = n + iw, where
then h(s) becomes
1 h(s) =
( S  n  iw)(s  n + iw)
212 Chapter 7. Response to Actuation of the ControlsOpen Loop
Figure 7.5 Admittances of a firstorder system.
and the inverse is found from item 13. Table A.l to be
1 h(t) =  en' sin wt
w
For a stable system n is negative and (7.3,ll) describes a damped sinusoid of fre quency w. This is plotted for various in Fig. 7.6. Note that the coordinates are so chosen as to lead to a oneparameter family of curves. Actually the above result only applies for l 5 1. The corresponding expression for 2 1 is easily found by the same method and is
1 h(t) = 7 en' sinh w't (7.3,12)
w
where
Graphs of (7.3,12) are also included in Fig. 7.6, although in this case the secondor der representation could be replaced by two firstorder elements in series.
7.4 StepFunction Response 213
Figure 7.6 Impulsive admittance of secondorder systems.
7.4 StepFunction Response
This is like the impulse response treated above except that the input is the unit step function I( t) , with transform 11s (Table A.l). The response in this case is called the step response or indicia1 admittance, and is denoted d,,(t). It follows then that
Since the initial values (at t = 0) of h,(t) and .dij(t) are both zero, the theorem (A.2,4) shows that
(a) .
Thus dij(t) can be found either by direct inversion of (7.4,lb) or by integration of hij(t). By either method the results for first and secondorder systems are readily ob tained, and are as follows (for a single inputlresponse pair the subscript is dropped):
214 Chapter 7. Response to Actuation of the ControlsOpen Loop
ont/2r
Figure 7.7 Indicia1 admittance of secondorder systems.
Firstorder system:
d(t) = T(l  eCn)
Secondorder system:
n 1  en'(cos wt   sin wt) , 5 < 1 w I (7.494)
For 5 > 1, see Appendix A.2. Graphs of the indicia1 responses are given in Figs. 7.5b and 7.7. The asymptotic value of d ( t ) as t , is called the static gain K. Applying the
final value theorem (A.2,12) (7.4,l) yields
Thus
lirn d ( t ) = lim sd ( s ) = lim G(s) tm s0 s0
K = lim G(s) s0
7.5 Frequency Response
When a stable linearlinvariant system has a sinusoidal input, then after some time the transients associated with the starting conditions die out, and there remains only a steadystate sinusoidal response at the same frequency as that of the input. Its ampli tude and phase are generally different from those of the input, however, and the ex pression of these differences is embodied in the frequencyresponse function.
7.5 Frequency Response 215
Consider a single inputlresponse pair, and let the input be the sinusoid a , cos ot . We find it convenient to replace this by the complex expression c = Alei"', of which a , cos wt is the real part. A , is known as the complex amplitude of the wave. The re sponse sinusoid can be represented by a similar expression, x = A,e'"', the real part of which is the physical response. As usual, x and c are interpreted as rotating vectors whose projections on the real axis give the relevant physical variables (see Fig. 7 .8~) .
From Table A. 1, item 8, the transform of c is
(6)
Figure 7.8 (a) Complex input and response. (b) Effect of singularity close to axis.
216 Chapter 7. Response to Actuation of the ControlsOpen Loop
Therefore
The function G(s) is given by (7.2,9) so that
The roots of the denominator of the r.h.s. are
A, A,, iw
so that the application of the expansion theorem (A.2,10) yields the complex output
Since we have stipulated that the system is stable, all the roots A, .. A, of the charac teristic equation have negative real parts. Therefore e"" + 0 as t + for r = 1 n, and the steadystate periodic solution is
Thus
A, = A,G(iw)
is the complex amplitude of the output, or
the frequency response function, is the ratio of the complex amplitudes. In general, G(io) is a complex number, varying with the circular frequency w. Let it be given in polar form by
G(iw) = K M ~ ' ~ (7.5,6)
where K is the static gain (7.45). Then
From (7.5,7) we see that the amplitude ratio of the steadystate output to the input is ~A2/A1I = KM: that is, that the output amplitude is a, = KMa,, and that the phase re
7.5 Frequency Response 217
oT=
Locus of Me@ (semicircle)
wTc 1
Figure 7.9 Vector plot of Me'' for firstorder systems.
lation is as shown on Fig. 7 . 8 ~ . The output leads the input by the angle cp. The quan tity M, which is the modulus of G(iw) divided by K, we call the magnification factor, or dynamic gain, and the product KM we call the total gain. It is important to note that M and cp are frequencydependent.
Graphical representations of the frequency response commonly take the form of either vector plots of Meiw (Nyquist diagram) or plots of M and cp as functions of fre quency (Bode diagram). Examples of these are shown in Figs. 7.9 to 7.13.
EFFECT OF POLES AND ZEROS ON FREQUENCY RESPONSE
We have seen (7.2,9) that the transfer function of a linearlinvariant system is a ratio of two polynomials in s, the denominator being the characteristic polynomial. The roots of the characteristic equation are the poles of the transfer function, and the roots of the numerator polynomial are its zeros. Whenever a pair of complex poles or zeros lies close to the imaginary axis, a characteristic peak or valley occurs in the ampli tude of the frequencyresponse curve together with a rapid change of phase angle at the corresponding value of w. Several examples of this phenomenon are to be seen in the frequency response curves in Figs. 7.14 to 7.18. The reason for this behavior is readily appreciated by putting (7.2,9) in the following form:
(s  21) . (s  22) ... (s  z,,) G(s) =
(S  Al) . (S  A2) (S  A,)
where the hi are the characteristic roots (poles) and the zi are the zeros of G(s). Let
where p, r, a, p are the distances and angles shown in Fig. 7.8b for a point s = iw on the imaginary axis. Then
218 Chapter 7. Response to Actuation of the ControlsOpen Loop
oT (b)
Figure 7.10 Frequencyresponse curvesfirstorder system.
When the singularity is close to the axis, with imaginary coordinate w' as illustrated for point S on Fig. 7.8b, we see that as w increases through w' , a sharp minimum oc curs in p or r, as the case may be, and the angle cr or P increases rapidly through ap proximately 180". Thus we have the following cases:
1. For a pole, in the left halfplane, there results a peak in / G I and a reduction in cp of about 180".
2. For a zero in the left halfplane, there is a valley in / G I and an increase in cp of about 180".
3. For a zero in the right halfplane, there is a valley in I G I and a decrease in cp of about 180".
7.5 Frequency Response 219
Figure 7.11 Vector plot of Me" for secondorder system. Damping ratio [ = 0.4
FREQUENCY RESPONSE OF FIRSTORDER SYSTEM
The firstorder transfer function, written in terms of the time constant T is
whence
K = lim G(s) = T s0
@I@, Figure 7.12 Frequencyresponse curvessecondorder system.
222 Chapter 7. Response to Actuation of the ControlsOpen Loop
lo3     phugold; Short I  I period 
I  I I I
 I I I
I I I
lo2  I I I I 1 ! 1 1 1 I I l l 9
Figure 7.14 Frequencyresponse functions, elevator angle input. Jet transport cruising at high altitudes. (a) Speed amplitude. (b) Speed phase.
The frequency response is determined by the vector G(iw)
whence
7.5 Frequency Response 223
Figure 7.15 Frequencyresponse functions, elevator angle input. Jet transport cruising at high altitude. (a) Angle of attack amplitude. (b) Angle of attack phase.
From (7.5,9), M and cp are found to be
A vector plot of Meiq is shown in Fig. 7.9. This kind of diagram is sometimes called the transferfunction locus. Plots of M and cp are given in Figs. 7 . 1 0 ~ and 6. The ab scissa is f T or log wT where f = w/2.rr, the input frequency. This is the only parame
224 Chapter 7. Response to Actuation of the ControlsOpen Loop
I I I
I 1 1 1 1 1 1
1 o  ~ 1 0' 1 o0 10' w (radls)
(b )
Figure 7.16 Frequencyresponse functions, elevator angle input. Jet transport cruising at high altitude. (a) Pitchrate amplitude. (b) Pitchrate phase.
ter of the equations, and so the curves are applicable to all firstorder systems. It should be noted that at w = 0, M = 1 and cp = 0. This is always true because of the definitions of K and G(s)it can be seen from (7.4,5) that G(0) = K.
FREQUENCY RESPONSE OF A SECONDORDER SYSTEM
The transfer function of a secondorder system is given in (7.3,9). The frequencyre sponse vector is therefore
7.5 Frequency Response 225
(b)
Figure 7.17 Frequencyresponse functions, elevator angle input. Jet transport cruising at high altitude. (a ) Flightpath angle amplitude. (b) Flightpath angle phase.
From the modulus and argument of (7.5,l l), we find that
226 Chapter 7. Response to Actuation of the ControlsOpen Loop
1 02 1 0' 1 oO 10' o (radls)
(b )
Figure 7.18 Frequencyresponse functions, elevator angle input. Jet transport cruising at high altitude. (a ) Load factor amplitude. (b) Load factor phase.
A representative vector plot of Me'", for damping ratio 6 = 0.4, is shown in Fig. 7.1 1, and families of M and cp are shown in Figs. 7.12 and 7.13. Whereas a single pair of curves serves to define the frequency response of all firstorder systems (Fig. 7. lo), it takes two families of curves, with the damping ratio as parameter, to display the char acteristics of all secondorder systems. The importance of the damping as a parame ter should be noted. It is especially powerful in controlling the magnitude of the reso nance peak which occurs near unity frequency ratio. At this frequency the phase lag is by contrast independent of 6, as all the curves pass through cp = 90" there. For all
7.5 Frequency Response 227
values of (, M + 1 and cp + 0 as w/wn + 0. This shows that, whenever a system is driven by an oscillatory input whose frequency is low compared to the undamped natural frequency, the response will be quasistatic. That is, at each instant, the output will be the same as though the instantaneous value of the input were applied stati cally.
The behavior of the output when 5 is near 0.7 is interesting. For this value of 5, it is seen that cp is very nearly linear with wlw, up to 1.0. Now the phase lag can be in terpreted as a time lag, T = (cp/2n)T = cplw where T is the period. The output wave form will have its peaks retarded by r sec relative to the input. For the value of 5 un der consideration, cpl(wlwn) = d 2 or cplw = d 2 w n = iT,,, where Tn = 2rr1wn, the un damped natural period. Hence we find that, for 5 = 0.7, there is a nearly constant time lag r = 4Tn, independent of the input frequency, for frequencies below reso nance.
The "chain" concept of higherorder systems is especially helpful in relation to frequency response. It is evident that the phase changes through the individual ele ments are simply additive, so that higherorder systems tend to be characterized by greater phase lags than loworder ones. Also the individual amplitude ratios of the el ements are multiplied to form the overall ratio. More explicitly, let
be the overall transfer function of n elements. Then
so that
On logarithmic plots (Bode diagrams) we note that
log KM = 2 log KrMr r= l
Thus the log of the overall gain is obtained as a sum of the logs of the component gains, and this fact, together with the companion result for phase angle (7.5,13) greatly facilitates graphical methods of analysis and system design.
RELATION BETWEEN IMPULSE RESPONSE AND FREQUENCY RESPONSE
We saw earlier (7.3,4), that h(t) is the inverse Fourier transform of G(iw), which we can now identify as the frequency response vector. The reciprocal Fourier transform relation then gives
228 Chapter 7. Response to Actuation of the ControlsOpen Loop
that is, the frequency response and impulsive admittance are a Fourier transform pail:
7.6 Longitudinal Response
To treat this case we need the matrices A and B of (7.1,4). A is given in (4.9,18), but we have not yet given B explicitly. On the right side of (4.9,18) we have the product BAc given by
BAc = (m  Z,b) AM, M, AZ,  + 
I, I, (m  Z*) 0
We now have to specify the control vector c and the corresponding aerodynamic forces and moment. For longitudinal control, we assume here that the available con trols are well enough represented by
and that the incremental aerodynamic forces and moment that result from their actua tion are given by a set of control derivatives X,= and so on, in the form
Additional elements can be added to c and to (7.6,3) if the situation requires it. The use of constant derivatives, as in (7.6,3), to describe the force output of the
propulsion system in response to throttle input does not allow for any time lag in the buildup of engine thrust since it implies that the thrust is instantaneously proportional to the throttle position. This is not unreasonable for propeller airplanes, but it is not a good model for jets in situations when the shortterm response is important, as for ex ample in a balked landing. To allow for this effect when the system is modeled in the Laplace domain, one can use control transfer functions instead of control derivatives. That is, one can replace, for example, X$ by G,,p(s). If the system model is in the time domain, the same result can be obtained by adding an additional differential equation and an additional variable. This latter method is illustrated in the example of Sec. 8.5.
By substituting (7.6,3) into (7.6,l) we derive the matrix B to be
7.7 Responses to Elevator and Throttle 229
With A and B known, we can compute the desired transfer functions and responses. (In this example the elevator angle is in radians, and English units are used for all other quantities). We calculate responses for the same jet transport as was used previ ously in Sec. 6.2, with A given by (6.2,l). The nondimensional elevator derivatives are :
from which the dimensional derivatives are calculated as
Xae = C,, kPu:S = 3.717
z = C z,c, 1 zpu:S = 3.551 X 105
Ma,, = C ,,,, $pu$S~ = 3.839 X 10'
For the throttle, we arbitrarily choose a value of X,jm = 0.3 g when 6, = 1, and Zsp and Msp = 0. With these values we get for the matrix B:
7.7 Responses to Elevator and Throttle
RESPONSE TO ELEVATOR
When the only input is the elevator angle AS, the system reduces to
where bij are the elements of B. Solving for the ratios G,&) = Aii/As, and so on yields the four transfer functions.' Each is of the form (7.2,8). For this case the char
'In the subscripts for the transfer function symbols, the symbol A is omitted in the interest of sim plicity.
230 Chapter 7. Response to Actuation ofthe ControlsOpen Loop
acteristic polynomial is the left side of (6.2,6) (with s replacing A), and the numerator polynomials are
There are two other response quantities of interest, the flight path angle y and the load factor n,. Since 8, = 0, A8 = 8, and Ay = A8  Aa, it readily follows that
G,, = Go,  G,, (7.793)
We define n, to be (see also Sec. 3.1)
It is equal to unity in horizontal steady flight, and its incremental value during the re sponse to elevator input is
An, =  AZIW =  (ZuAu + ZWw + Z,q + Z,W + Z,eAG,)lW
After taking the Laplace transform of the preceding equation and dividing by AG,, we get the transfer function for load factor to be
The total gain and phase of the frequency responses calculated by (7.5,6) from five of the above six transfer functions (for Au, w, q, A y, and An,) are shown in Figs. 7.14 to 7.18.
The exact solutions show that the responses in the "trajectory" variables u and y are dominated entirely by the large peak at the lowfrequency Phugoid mode. Be cause of the light damping in this mode, the resonant gains are very large. The peak IGUaCI of nearly 3 X lo4 means that a speed amplitude of 100 fps would result from an elevator angle amplitude of about 100/(3 X lo4) rad, or about 0.2". Similarly, at reso nance an amplitude of 10" in y would be produced by an elevator amplitude of about Q". For both of these variables the response diminishes rapidly with increasing fre quency, becoming negligibly small above the shortperiod frequency. The phase an gle for u, Fig. 7.14b, is zero at low frequency, decreases rapidly to near  180" at the phugoid frequency (very much like the lightly damped secondorder systems of Fig. 7.13) and subsequently at the shortperiod frequency undergoes a second drop char acteristic of a heavily damped secondorder system. The "chain" concept of highor der systems (Sec. 7.2) is well exemplified by this graph.
By contrast, the attitude variables w and q show important effects at both low and high frequencies. The complicated behavior of w near the phugoid frequency indi cates the sort of thing that can happen with highorder systems. It is associated with a polelzero pair of the transfer function being close to one another. Again, above the shortperiod frequency, the amplitudes of both w and q fall off rapidly.
The amplitude of the load factor An, has a very large resonant peak at the phugoid frequency, almost 1001rad. It would not take a very large elevator amplitude at this frequency to cause structural failure of the wing!
7.7 Responses to Elevator and Throttle 231
StepFunction Response
The response of the airplane to a sudden movement of the elevator is shown by the step response. This requires a solution in the time domain as distinct from the pre ceding solution, which was in the frequency domain. Time domain solutions are com monly obtained simply by integrating (7.1,4) by a RungeKutta, Euler, or other inte gration scheme, the choice being dependent on the order of the system, accuracy required, computer available, and so on. The software used2 for the example to follow does not integrate the equations, but instead uses an alternative method. It inverts the transfer function using the Heavyside expansion theorem (A.2,10). For the same jet airplane and flight condition as in the preceding example, the control vector for ele vator input is c = [A6, OIT and A and B are as before. (Note that only the first col umn of B is needed.) Time traces of speed, angle of attack, and flight path angle are shown in Figs. 7.19 and 7.20 for two time ranges when the elevator displacement is one degree positive, that is, down.
It is seen from Fig. 7.19, which shows the response during the first 10 sec, that only the angle of attack responds quickly to the elevator motion, and that its variation is dominated by the rapid, welldamped shortperiod mode. By contrast, the trajec tory variables, speed, and flight path angle, respond much more slowly. Figure 7.20, which displays a 10 min time span, shows that the dynamic response persists for a very long time, and that after the first few seconds it is primarily the phugoid mode that is evident.
The steady state that is approached so slowly has a slightly higher speed and a slightly smaller angle of attack than the original flight conditionboth changes that would be expected from a down movement of the elevator. The flight path angle is seen to be almost unchangedit increased by about onetenth of a degree. The reason for an increase instead of the decrease that would be expected in normal cruising flight is that at this flight condition the airplane is flying below its minimumdrag speed.
If the reason for moving the elevator is to establish a new steadystate flight con dition, then this control action can hardly be viewed as successful. The long lightly damped oscillation has seriously interfered with it. Clearly, longitudinal control, whether by a human or an automatic pilot, demands a more sophisticated control ac tivity than simply moving it to its new position. We return to this topic in Chap. 8.
Phugoid Approximation
We can get an approximation to the transfer functions by using the phugoid ap proximation of Sec. 6.3. The differential equation is (6.3,6) with control terms added, that is,
'Program CC, see Appendix A.7
232 Chapter 7. Response to Actuation of the ControlsOpen Loop
Time, s
(a) Speed
( b ) Angle of attack
. 1 I I I I 0 2 4 6 8
Time, s (c ) Fl~ght path angle
Figure 7.19 Response to elevator (A6, = lo). Jet transport cruising at high altitude.
.01
D m t( d
.02
.03


I I I I 0 12 24 36 48 60
Time, s
7.7 Responses to Elevator and Throttle 233
90
(a ) Speed
Ttrne, s
( b ) Angle of attack
Ay,, = 0.0024 rad
V I I I I
( c ) Fl~ght path angle
Figure 7.20 Response to elevator (A6, = 1 O ) . Jet transport cruising at high altitude.
234 Chapter 7. Response to Actuation of the ControlsOpen Loop
where A is the matrix of (6.3,6). After taking the Laplace transform of this equation and solving for the ratios of the variables, we find the transfer functions to be
where f(s) is the characteristic polynomial of (6.3,8),
f(s) = AS^ + Bs + C
and
ShortPeriod Approximation
We can also get useful approximations to the transfer functions for 8, q, and a by using the shortperiod approximation of Sec. 6.3. Instead of (7.7,l) we get the equa tion
(SI  A) [ :] = B AS. (7.7,9)
in which A is the matrix of (6.3,13) and B is obtained from (7.6,4) as
7.7 Responses to Elevator and Throttle 235
When calculating the transfer functions we take note of the fact that ij = she. After solving (7.7,9) for the appropriate ratios we get the results
where
M , M" Zae a, = u,, 
  
I, 1, m
The frequency responses calculated with the above approximate transfer func tions are shown on Figs. 7.15 to 7.18. It is seen that the phugoid approximation is ex act at very low frequencies and the shortperiod approximation is exact in the high frequency limit. For frequencies between those of the phugoid and shortperiod modes, one approximation or the other can give reasonable results.
RESPONSE TO THE THROTTLE
For the same jet airplane and flight condition as in previous numerical examples, we calculate the response to a step input in the throttle of A6, = i, which corresponds to a thrust increment of 0.05W. The matrix B is given by (7.63) and A c is
A c = [ O 1/6IT (7.7,13)
The numerical results are shown in Fig. 7.21. Because the model has not included any engine dynamics, the results are not valid for the first few seconds. However, this region is not of much interest in this case. The motion is clearly seen to be dominated by the lightly damped phugoid. The speed begins to increase immediately, before the other variables have time to change. It then undergoes a slow damped oscillation, ul timately returning to its initial value. The angle of attack varies only slightly, and y makes an oscillatory approach to its final positive value Y,,~. The new steady state is a climb with Au = A a = 0. When the thrust line does not pass through the CG, the re sponse is different in several details. Principally, the moment of the thrust causes a
236 Chapter 7. Response to Actuation of the ControlsOpen Loop
40 I I I I 0
1 100
1 200 300 400 500 600
Time, s
(a) Speed
.05 I I I I 0
I 100 200 300 400 500 600
T~me, s (b ) Angle of attack
y,, = 0105 rad I I
Time, s
(c ) Flight path angle
Figure 7.21 Response to throttle. Jet transport cruising at high altitude. Thrust line passing through CG.
7.8 Lateral Steady States 237
rapid change in a , followed by an oscillatory decay to a new Aa,, # 0, and the speed converges to a new value Au,, # 0.
7.8 Lateral Steady States
The basic flight condition is steady symmetric flight, in which all the lateral variables p, p, r, 4 are identically zero. Unlike the elevator and the throttle, the lateral controls, the aileron and rudder, are not used individually to produce changes in the steady state. This is because the steady state values of P, p, r, 4 that result from a constant 6, or 6, are not generally of interest as a useful flight condition. There are two lateral steady states that are of interest, however, each of which requires the joint application of aileron and rudder. These are the steady sideslip, in which the flight path is recti linear, and the steady turn, in which the angular velocity vector is vertical. We look into these below before proceeding to the study of dynamic response to the lateral controls.
THE STEADY SIDESLIP
The steady sideslip is a condition of nonsymmetric rectilinear translation. It is some times used, particularly with light airplanes, to correct for crosswind on landing ap proaches. Glider pilots also use this maneuver to steepen the glide path, since the L/D ratio decreases due to increased drag at large P. In this flight condition all the time derivatives in the equations of motion (except iE) and the three rotation rates p, q, r are zero. It is simplest in this case to go back to (4.9,2)(4.9,6), from which we derive the following:
We use (4.9,17) for aerodynamic forces, and for the control forces use the following as a reasonable representation:
We now add the assumption that 8, is a small angle and get the resulting equation
In this form, u is treated as an arbitrary input, and (a,, a,, 4) as outputs. (See Exercise 7.6.) Clearly, there is an infinity of possible sideslips, since v can be chosen arbitrar ily. Note that the other three variables are all proportional to v. We illustrate the steady sideslip with a small general aviation airplane' of 30ft (9.14 m) span and a gross weight of 2400 lb (10,675 N). The altitude is sea level and C, = 1.0, corre
238 Chapter 7. Response to Actuation of the ControlsOpen Loop
Table 7.2 Nondimensional DerivativesGeneral Aviation Airplane (expressed in rad' and (rad/s)')
sponding to a speed of 112.3 fps (34.23 rn/s), and the wing area is 160 ft2 (14.9 m2). The nondimensional derivatives are given in Table 7.2, from which the numerical system equation is found from (7.8,3) to be (see Exercise 7.5)
It is convenient to express the sideslip as an angle instead of a velocity. To do so we recall that p ulu,, with u, given above as 112.3 fps. The solution of (7.8,4) is found to be
We see that a positive sideslip (to the right) of say 10" would entail left rudder of 3" and right aileron of 29.6". Clearly the main control action is the aileron displacement, without which the airplane would, as a result of the sideslip to the right, roll to the left. The bank angle is seen to be only lo to the right so the sideslip is almost flat.
THE STEADY TURN
We define a "truly banked" turn to be one in which (1) the vehicle angular velocity vector w is constant and vertical (see Fig. 7.22) and (2) the resultant of gravity and centrifugal force at the mass center lies in the plane of symmetry (see Fig. 7.23). This corresponds to flying the turn on the turnandbank indi~ator .~ It is quite common for turns to be made at bank angles that are too large for linearization of sin 4 and cos 4 to be acceptable, although all the state variables other than C$ and V are small. Thus we turn to the basic nonlinear equations in Sec. 4.7 for this analysis. The large bank angle has the consequence that coupling of the lateral and longitudinal equations oc curs, since more lift is needed to balance gravity than in level flight. Thus not only the aileron and rudder but the elevator as well must be used for turning at large 4.
3 ~ a s e d on the Piper Cherokee. The control derivatives were taken from McCormick (1979). We es timated the stability derivatives. The numerical values used may not truly represent this airplane.
4Neglecting the fact that the pilot and indicator are not right at the CG.
7.8 Lateral Steady States 239
,Horizontal plane   11
z\F\I b ~ e r t i c a l axis of turn
Figure 7.22 Steady climbing turn.
xz plane / Figure 7.23 Gravity and acceleration in turn.
240 Chapter 7. Response to Actuation of the ControlsOpen Loop
The bodyaxis angular rates are given by
which for small elevation angle 13 yields
We now apply the second condition for a trulybanked turn, that is, that the ball be centered in the turnandbank indicator. This means that the vector mg  ma,, where a, is the acceleration vector of the CG, shall have no y component. But ma, is the resultant external force f, so that from (4.5,6)
where A is the resultant aerodynamic force vector. Thus we conclude that the aerody namic force must lie in the xz plane, and hence that Y = 0. We consider the case when there is no wind, so that
and choose the body axes so that a, = w = 0. We now use (4.7,l) with all the vari ables constant and only u and 4 not small to get:
Y =  m g s i n + + m r u = O
Z = mg cos 4  mqu
When u is small, a reasonable assumption for a truly banked turn, we also have that u = V, the flight speed. It follows from (7.8,7a) that
rV sin 4 = 
g
and with the value of r obtained from (7.8,6)
wv tan 4 = 
The load factor n, is obtained from (7.8,7b):
z q v n , =   = c o s + + 
mg g
With q from (7.8,6) this becomes
Vw sin 4 nz = COS 4 +
g
By using (7.8,8) to eliminate Vw we get
n, = sec 4
7.8 Lateral Steady States 241
We note from (5.1,l) that Z = L in this case, so that n = L/W = n,. The incremen tal lift coefficient, as compared with straight flight at the same speed and height, is
We can now write down the equations governing the control angles. From (4.7,2), to first order, L = M = N = 0, so we have the five aerodynamic conditions
c  c = c = c = o I  m n Y
and AC, = (n  l)Cw
On expanding these with the usual aerodynamic derivatives, we get
In these relations B , g , i are known from (7.8,6), that is,
The five equations (7.8,11) for the five unknowns [P, S,, S,] and [Aa, AS,] uncouple into two independent sets:
 
f i
4
i  
and
When (7.8,9) is used to eliminate + from (7.8,14), and after some routine algebra, the solution for AS, is found to be (see Exercise 7.6)
=
  b
 6  2 v  C
sin+ 2 v b
cos +  2 v
 
w
242 Chapter 7. Response to Actuation of the ControlsOpen Loop
Except for far forward CG positions and low speeds, the angles given by (7.8,15) are moderate. The similarity of this expression to that for elevator angle per g in a pull up (3.1,6a) should be noted. They are in fact the same in the limit n + m. The eleva tor angle per g in a turn is therefore not very different from that in a vertical pullup.
Finally, the lateral control angles are obtained from the solution of (7.8,13).
NUMERICAL EXAMPLE
The rudder and aileron angles in a steady trulybanked turn are calculated by way of example for the same general aviation airplane as was used above for the sideslip. The altitude is sea level, the speed is 125 fps (38.1 mls) and the stability and control derivatives are as in Table 7.2. The solution of (7.8,13) for climb angles between  10" and + 10" shows that the sideslip angle P remains less than 1.5" and the rudder and aileron angles are as shown in Figs. 7.24 and 7.25. The value of C, varies over the range of bank angles used from 0.8 to 1.6, so several of the stability derivatives are significantly affected. It is seen that the aileron angle is always positive for a right turnthat is, the right aileron is down (stick to the left), and that the rudder is usually negative (right rudder) although its sign may reverse in a steep climb. The strong ef fect of the climb angle derives from the fact that the roll rate p is proportional to 8 and changes sign with it. Thus the terms C,J and Cnp$ in the moment equations change sign between climbing and descending and affect the control angles required to produce zero moment.
1 0 I I I I I 0 10 20 30 40 50 60
Bank angle, go
Figure 7.24 Rudder angle in turn.
ui 2  ai  m
L.
5 a, u u 3 2  [r
4
General aviation airplane Speed 125 fps Sea level
10" climb ..... .... ..... ..._....
Horizontal 0  _____  \     
I OD descent ..\, 
\
'.
7.9 Lateral Frequency Response 243
Bank angle,
Figure 7.25 Aileron angle in turn.
General aviation airplane Speed 125 fps Sea level
I I loodescent
 I I I
 I I I I
 I I
/ Horizontal 




I I 0 10 20 30 40 50 60 70 80
Finally, it may be remarked that the control angles obtained would have been substantially different had it been stipulated that P, not C,, should be zero in the turn. It would not then be possible, however, to satisfy the requirement that the ball be cen tered in the turnandback indicator.
90
7.9 Lateral Frequency Response
The procedure for calculating the response of the airplane to sinusoidal movement of the rudder or aileron is similar to that used for longitudinal response in Sec. 7.6. The
Table 7.3 Control DerivativesB747 Jet Transport (expressed in rad')
C"
 1.973 X
0.1257
CI
 1.368 X lo'
6.976 X
4,
8,
CY
0.1 146
244 Chapter 7. Response to Actuation of the ControlsOpen Loop
aerodynamics associated with the two lateral controls are given by a set of control de rivatives:
The Laplace transform of the system equation (4.9,19) is then
where B is
NUMERICAL EXAMPLE
For our numerical example we use the same jet transport and flight condition as in Sec. 7.6. A is given by (6.7,l) and the control derivatives are given in Table 7.3, from which, with the definitions of Table 7.1, the elements of B are calculated to be
The eight transfer functions are then as in (7.2,8), where f(s) is the characteristic polynomial of (6.7,2) (with s instead of A) and with the numerators as follows:
Nu," = 2.896s2 + 6.542s + 0.6220 (a)
Nu,r = 5.642s3 + 379.4s2 + 167.9s  5.934 (b) NPaa = 0.1431s3 + 0.02730s2 + 0.1 102s (c) NPar = 0.1 144s3  0.1997s2  1.368s (4 N,," = 0.003741s3  0.002708s2  0.0001394s + 0.004539 (e ) (7.93) Nr,r = 0.4859s3  0.2327s2  0.009018s  0.05647 ( f )
N+8a = .1431s2 + 0.02730s + 0.1 102 (8) N+,r = 0.1 144s2  0.1997s  1.368 (h)
7.9 Lateral Frequency Response 245
From the transfer functions Gi,(s) = N,(s)lf(s), the frequency response functions Gij(iw) were calculated for both aileron and rudder inputs. The results for u , 4 and r are shown on Figs. 7.26 and 7.27. The most significant feature in all of these re sponses is the peak in the amplitude at the Dutch Roll frequency, and the associated sharp drop in phase angle.
At zero frequency we see from (7.9,5c and d) that the roll rate amplitude is zero for both inputs. All the other variables have finite values at w = 0. Even for moderate
1 80 1 o  ~ lo' 1 oO 10'
w (radls)
(b)
Figure 7.26 Frequencyresponse functions, rudder angle input. Jet transport cruising at high altitude. (a) Sideslip amplitude. (b) Sideslip phase. (c ) Roll amplitude. (d) Roll phase. ( e ) Yawrate amplitude. ( f ) Yawrate phase.
246 Chapter 7. Response to Actuation of the ControlsOpen Loop
o (radls)
(dl
Figure 7.26 (Continued)
control angles, however, the steadystate values of P = ulu, and 4 are very large (see Exercise 7.10). Hence the linearity assumption severely constrains these zero fre quency solutions. If, however, we postulate that the control angles are so small that the linearity conditions are met, then there is a steady state with constant values of 4, P, and r. This can only be a horizontal turn in which the angular velocity vector is
0, = [O 4 s s LIT where q,, = fl sin 4
r,y:,, = fl cos 4
7.10 Approximate Lateral Transfer Functions 247
7.1 0 Approximate Lateral Transfer Functions
Approximate transfer functions that can be written out explicitly, and that reveal the main aerodynamic influences in a particular frequency range, can be very useful in designing control systems. In Sec. 6.8 we presented two approximate secondorder systems that simulate the complete fourthorder system insofar as the characteristic modes are concerned. These same approximations can be used to get approximate transfer functions for control response.
248 Chapter 7. Response to Actuation of the ControlsOpen Loop
o (rad/s) (a)
Figure 7.27 Frequencyresponse functions, aileron angle input. Jet transport cruising at high altitude. (a) Sideslip amplitude. (b) Sideslip phase. (c) Roll amplitude. (d) Roll phase. (e) Yawrate amplitude. Cf) Yawrate phase.
SPIRALmOLL APPROXIMATION
When aerodynamic control terms are added to (6.8,9) and the Laplace transform is taken, the result is
7.10 Approximate Lateral Transfer Functions 249
t I roll 11 I uutcn
1
w (radls)
(d) Figure 7.27 (Continued)
In (7.10,l) the B and N derivatives are as defined in Sec. 6.8, and the 3, derivatives are
where 6 is either 6, or 6,. From (7.10,l) we get the desired transfer functions. The de nominators are all the same, obtained from (6.8,11) as
Cs2 + DS + E (7.10,3)
250 Chapter 7. Response to Actuation of the ControlsOpen Loop
o (radls)
I f )
Figure 7.27 (Continued)
and the numerators are (again using 6 for either 6, or 6,):
7.10 Approximate Lateral Transfer Functions 251
The coefficients in these relations are:
a,=%,; a , =  % , ( 2 , + N r )  u , N ,
a , = %,(2,N,.  2,N,J  u,,(2,Np  X,N,) + 2,g
ao = g ( y r J f ,  %Nr> (7.105)
6 , = 3 8 2 , ; bo = ~,)(3,Su",  =YuK6) + % 8 ( 3 U N ,  2 , N U )
d2 = %,Nu; dl = 9 , (2 ,N ,  2 ,Nu) ; do = g(%,N,  2 , N 8 )
DUTCH ROLL APPROXIMATION
Following the analysis of Sec. 6.8 and adding control terms to the aerodynamics the reduced system equations are
ir = 9 , v  uor + A%,. i. = Nuu + Nrr + AN,
From (7.10,6) we derive the canonical equation
x = A x + Bc
where x = [ u rIT; c=[8' , SrlT
and where
With the system matrices given by (7.10,7) the approximate transfer functions are found in the form of (7.2,8) (see Exercise 7.7) with
and
The accuracy of the preceding approximations is illustrated for the example jet trans port on Figs. 7.26 and 7.27. Two general observations can be made: ( I ) The Dutch Roll approximation is exact in the limit of high frequency, and ( 2 ) the spirallroll ap proximation is exact as o + 0. In this respect the situation is entirely analogous to that of the longitudinal case, with the spirallroll corresponding to the phugoid, and the Dutch Roll to the shortperiod mode. There are ranges of frequency in the middle where neither approximation is good. We repeat that lateral approximations must be used with caution, and that only the exact equations can be relied on to give accurate results.
252 Chapter 7. Response to Actuation of the ControlsOpen Loop
7.11 Transient Response to Aileron and Rudder
We have seen that useful lateral steady states are produced only by certain definite combinations of the control deflections. It is evident then that our interest in the re sponse to a single lateral control should be focused primarily on the initial behavior. The equations of motion provide some insight on this question directly. Following a step input of one of the two controls the state variables at t = 0+ are all still zero, and from (4.9,19) we can deduce that their initial rates of change are related to the control angles by
The initial sideslip rate i, is thus seen to be governed solely by the rudder and, since 9, > 0, is seen to be positive (slip to the right) when 6, is positive (left rud der). Of somewhat more interest is the rotation generated. The initial angular acceler ation is the vector
The direction of this vector is the initial axis of rotation, and this is of interest. It lies in the xz plane, the plane of symmetry of the airplane, as illustrated in Fig. 7.28~. The angle 5 it makes with the x axis is, of course,
Let us consider the case of "pure" controls, that is, those with no aerodynamic cross coupling, so that Lg = Nsa = 0. The ailerons then produce pure rolling moment and
FC Principal axes
yaw control
i (b)
Figure 7.28 Initial response to lateral control. (a) General. (b) Example jet transport.
7.1 1 Transient Response to Aileron and Rudder 253
Figure 7.29 Angle of axis of rotation.
the rudder produces pure yawing moment. In that case we get for 6, = 0 the angle tR for response to rudder from
and similarly for response to aileron:
tan tA = lullz (7.1 1 s )
The angles tA, cR are seen to depend very much on the product of inertia I,. When it is zero, the result is as intuitively expected, the rotation that develops is about either the x axis (aileron deflected) or the z axis (rudder deflected). For a vehicle such as the jet transport of previous examples, with Ixp = 0.41z,,, the values of I,, I,, I, given by (4.5,11) yield the results shown in Fig. 7.29. The relations are also shown to scale in Fig. 7.28b for E = 20" (high angle of attack). It can be seen that there is a tendency for the vehicle to rotate about the principal x axis, rather than about the axis of the aerodynamic moment. This is simply because I,lI, is appreciably less than unity. Now the jet transport of our example is by no means "slender," in that it is of large span and has wingmounted engines. For an SST or a slender missile, the trend shown is much accentuated, until in the limit as aspect ratio + 0, both tan 6, and tan 5, tend to tan E , and the vehicle rotates initially about the x, axis no matter what control is used!
SOLUTION FOR LARGE ANGLES
The preceding analysis shows how a lateral response starts, but not how it continues. For that we need solutions to the governing differential equations. As remarked ear
254 Chapter 7. Response to Actuation of the ControlsOpen Loop
lier, at the beginning of this chapter, control responses can rapidly build up large val ues of some variables, invalidating the linear equations that we have used so far. There is a compromise available that includes only some nonlinear effects that is use ful for transport and general aviation airplanes, which are not subjected to violent maneuvers. The compromise is to retain a linear representation of the inertia and aerodynamic effects, but to put in an exact representation of the gravity forces. This allows the angles 4 , 8, and + to take on any values. As we shall see in the following example, the solution obtained is then limited by the airplane speed growing beyond the range of linear validity, that is, it is an aerodynamic nonlinearity that then con trols the useful range of the solution. When the procedure that led to (4.9,18 and 19) is repeated without the small angle approximations we get the following for 8, = 0 (see Exercise 7.8):
 g sin 19
cos 8 cos 4 )
AM
L7 1 p + (q sin 4 + i cos 4)tan 8 I I (q sin 4 + r cos 4)sec 8
The data for the B747 jet transport previously used was incorporated into the preced ing equations. A step aileron input of  15" was applied at time zero, the other con trols being kept fixed, and the solution was calculated using a fourthorder Runge Kutta algorithm. The results are shown in Fig. 7.30. The main feature is the rapid acquisition of roll rate, shown in Fig. 7.30b, and its integration into a steadily grow ing angle of bank (Fig. 7 . 3 0 ~ ) that reaches almost 90" in half a minute. Sideslip, yaw rate, and yaw angle all remain small throughout the time span shown. As the airplane rolls, with its lift remaining approximately equal to its weight, the vertical component of aerodynamic force rapidly diminishes, and a downward net force leads to negative 8 and an increase in speed. After 30 seconds, the speed has increased by about 10% of uo, and the linear aerodynamics becomes increasingly inaccurate. The maxi mum rotation rate is p = .05 radls, which corresponds to = 0.01. This is small enough that the neglect of the nonlinear inertia terms in the equations of motion is justified.
Time, s (a) Veloc~ty components
Time, s
(b) Angular velocity components
40 I I I I I 0 5 10 15 20 25 30
Time, s
(c ) Attitude angles
Figure 7.30 Response of jet transport to aileron angle; 6, =  15". (a ) Velocity components. (b) Angular velocity components. (c) Attitude angles.
256 Chapter 7. Response to Actuation of the ControlsOpen Loop
7.12 Inertial Coupling in Rapid Maneuvers
We saw in the last section how to include nonlinear gravity effects in control re sponse and how such effects manifest themselves in the response of a relatively se date vehicle. Of the other two categories of nonlinearityaerodynamic and inertial little in a general way can be said about the first. Aerodynamic characteristics, especially for flexible vehicles at high subsonic Mach numbers, are too varied and complex to admit of useful generalizations. A very elaborate (and very costly!) aero dynamic model is required for full and accurate simulation or computation. Not so, however, for the second category of nonlinearity. There is a class of problems, all generically connected, known by names such as roll resonance, spinyaw coupling, inertia coupling, and so on (Heppe and Celinker, 1957; Phillips, 1948; Pinsker, 1958) that pertain to largeangle motions, or even violent instabilities, that can occur on missiles, launch vehicles, and slender aircraft performing rapid rolling maneuvers. These have their source in the pq and p r terms that occur in the pitching and yawing moment equations. A detailed analysis of these motions would take us beyond the scope of this text. Some is given in Etlun (1972), and much more is given in the cited references. One very important conclusion, due to Phillips (1948), is that there is a band of roll rates for airplanes within which the airplane is unstable. At lower roll rates, the usual stability criteria apply. At rates above the band the airplane is gyrosta bilized in the way a spinning shell or top is. The lower of the critical roll rates for a normally stable airplane is given approximately by the lesser of
If the roll rate in a maneuver approaches or exceeds this value the possibility of a dangerous instability exists.
7.13 Exercises
7.1 A = [av] is a (3x3) matrix. Demonstrate the statement made in the text with respect to the numerator of (7.2,7) by writing out in full the adjoint of (sI  A).
7.2 Use the convolution theorem (Appendix A.3) to obtain an alternative proof of the the orem for frequency response. That is, for a system with transfer function G(s) and in put ei" the response for t + co is G(iw)ei".
7.3 An airplane is flying at the speed V* for which the thrust curve is tangent to the drag curve (Fig. 7.1). The throttle is then suddenly advanced to produce a higher thrust curve, such as is seen in the figure. The pilot controls the elevator so as to maintain exactly horizontal flight, in which case the drag curve is as in the figure. The ultimate steady state is at either P or Q. What will govern which it will be?
7.4 In a test flight procedure, the airplane is brought to a condition of steady horizontal flight in quiet air. The elevator is then displaced rapidly through a small angle, held briefly, and then returned as rapidly to its original position. Assume that the resulting input can be treated as an impulse at t = 0 (see Sec. 7.3).
7.13 Exercises 257
(a) Use the short period approximation (7.7,llb) to the transfer function for 0 to de rive a time domain solution for qt). Express the solution in terms of n, w, b,, and
b,. (b) Assuming that 8 and t can be determined very accurately from the flight test data,
and hence that n and w can be determined precisely, suggest how the experimen tal data could be used to determine b,, b,, c,, and c,. Note that if a, and a , could likewise be determined accurately, then the six equations (7.7,12) could in princi ple be used to solve for the six aerodynamic derivatives on the right side of the equations.
7.5 Use the nondimensional derivatives of Table 7.2 to calculate the coefficients of the matrix equation (7.8,4).
7.6 (a) Reformulate the equations for the steady sideslip (7.8,3) to use #J as input and (u, 6,, 6,) as outputs.
(b) Derive (7.8,15)
7.7 Derive (7.10,lO)
7.8 Derive (7.1 1,6) and (7.1 1,7).
7.9 An additional vertical control surface (6.J is added above the fuselage of an airplane, near the CG. It is capable of providing a side force, accompanied by a rolling mo ment, given by
What condition must be satisfied if (a,, a,., 6,) are to generate specified (Y, L, N)?
7.10 (a) Using the numerical data for the B747 example (Sec. 7.9), calculate the static gains for each of the eight responses that correspond to (7.9,5)that is, the val ues of ~ ( i w ) , ~ for w = 0.
(b) Calculate the slopes of the highfrequency asymptotes for each of the eight fre quency response amplitudes (express result in decadesldecade).
(c) Assume that w = 0, that is, that a steady state exists in response to one of the controls being deflected, such that #J = 15". For each of the two controls aileron and ruddercalculate the control angle, the sideslip angle, and the yaw rate r.
7.11 The elevator of the B747 airplane is oscillated at a frequency a little below that of the shortperiod mode.
(a) Use the results given in Fig. 7.18 to estimate the amplitude of the load factor if the elevator amplitude is 2".
(b) What elevator amplitude would lift a passenger seated near the CG from the seat?
(c) What elevator amplitude would cause the load factor to reach the FAR Part 25 limit maneuvering value of 2.5?
258 Chapter 7. Response to Actuation of the ControlsOpen Loop
7.14 Additional Symbols Introduced in Chapter 7
response to a unit step input
response to a unit pulse input
static gain
L,lI: + ILN,
magnification factor
N,lIi + ILL,
Y,lm
Dirac's delta function
phase angle
C H A P T E R 8
Closed Loop Control
8.1 General Remarks
The development of closed loop control has been one of the major technological achievements of the twentieth century. This technology is a vital ingredient in count less industrial, commercial, and even domestic products. It is a central feature of air craft, spacecraft, and all robotics. Perhaps the earliest known example of this kind of control is the flyball governor that James Watt used in his steam engine in 1784 to regulate the speed of the engine. This was followed by automatic control of torpedoes in the nineteenth century (Bollay, 1951), and later by the dramatic demonstration of the gyroscopic autopilot by Sperry in 1910, highly relevant in the present context. Still later, and the precursor to the development of a general theoretical approach, was the application of negative feedback to improve radio amplifiers in the 1930s. The art of automatic control was quite advanced by the time of the landmark four teenth Wright Brothers lecture (Bollay, 1951)'. Most of what is now known as "clas sical" control theorythe work of Routh, Nyquist, Bode, Evans, and others was de scribed in that lecture. From that time on the marriage of control concepts with analogue and digital computation led to explosive growth in the sophistication of the technology and the ubiquity of its applications.
Although openloop responses of aircraft, of the kind studied in some depth in Chap. 7, are very revealing in bringing out inherent vehicle dynamics, they do not in themselves usually represent real operating conditions. Every phase of the flight of an airplane can be regarded as the accomplishment of a set taskthat is, flight on a specified trajectory. That trajectory may simply be a straight horizontal line traversed at constant speed, or it may be a turn, a transition from one symmetric flight path to another, a landing flare, following an ILS or navigation radio beacon, homing on a moving target, etc. All of these situations are characterized by a common feature, namely, the presence of a desired state, steady or transient, and of departures from it that are designated as errors. These errors are of course a consequence of the un steady nature of the real environment and of the imperfect nature of the physical sys tem comprising the vehicle, its instruments, its controls, and its guidance system (whether human or automatic). The correction of errors implies a knowledge of them, that is, of errormeasuring (or statemeasuring) devices, and the consequent actuation of the controls in such a manner as to reduce them. This is the case whether control is by human or by automatic pilot. In the former casethe human pilotthe state in formation sensed is a complicated blend of visual and motion cues, and instrument readings. The logic by which this information is converted into control action is only
'In 1951 most aeronautical engineers were using slide rules and had not heard of a transfer function!
260 Chapter 8. Closed Loop Control
imperfectly understood, but our knowledge of the physiological "mechanism" that in tervenes between logical output and control actuation is somewhat better. In the latter casethe automatic controlthe sensed information, the control logic, and the dy namics of the control components are usually well known, so that system perfor mance is in principle quite predictable. The process of using state information to gov ern the control inputs is known as closing the loop, and the resulting system as a closedloop control or feedback control. The terms regulator and servomechanism describe particular applications of the feedback principle. Figure 8.1 shows a general block diagram describing the feedback situation in a flight control system. This dia gram models a linear invariant system, which is of course an approximation to real nonlinear timevarying systems. The approximation is a very useful one, however, and is used extensively in the design and analysis of flight control systems. In the di agram the arrows show the direction of information flow; the lowercase symbols are vectors (i.e. column matrices), all functions of time; and the uppercase symbols are matrices (in general rectangular). The vectors have the following meanings:
r : reference, input or command signal, dimensions (pX 1)
z: feedback signal, dimensions (pX 1)
e: error, or actuating, signal, dimensions (pX 1)
c: control signal, dimensions (mX 1)
g: gust vector (describing atmospheric disturbances), dimensions (IX 1 ) x: airplane state vector, dimensions (nX 1)
y: output vector, dimensions ( q x 1)
n: sensor noise vector, dimensions (qX 1)
Of the above, x and c are the same state and control vectors used in previous chapters. r is the system input, which might come from the pilot's controller, from an external navigation or fire control system, or from some other source. It is the com mand that the airplane is required to follow. The signal e drives the system to make z
Y J(s)
airframe
z
n v
Figure 8.1 A general linear invariant flight control system.
8.1 General Remarks 261
follow r . It will be zero when z = r. The makeup of the output vector y is arbitrary, constructed to suit the requirements of the particular control objective. It could be as simple as just one element of x, for example. The feedback signal z is also at the dis cretion of the designer via the choice of feedback transfer function H(s). The choices made for D(s), E(s) and H(s) collectively determine how much the feedback signal differs from the state. With certain choices z can be made to be simply a subset of x, and it is then the state that is commanded to follow r .
The vector g describes the local state of motion of the atmosphere. This state may consist of either or both discrete gusts and random turbulence. It is threedimen sional and varies both in space and time. Its description is inevitably complex, and to go into it in depth here would take us beyond the scope of this text. For a more com plete discussion of g and its closely coupled companion G ' the student should con sult Etkin (1 972) and Etkin (198 1).
In real physical systems the state has to be measured by devices (sensors) such as, for example, gyroscopes and Pitot tubes, which are inevitably imperfect. This im perfection is commonly modeled by the noise vector n, usually treated as a random function of time.
The equations that correspond to the diagram are (recall that overbars represent Laplace transforms):
e = p  z (a)
r = J(s)e (b)
iz = G(s)C + G1(s)g (c> (8.1,1)
p = Diz + EC (4 z = H(s)(g + ii) (el
In the time domain (8.1,l c) appears as
It follows that
G(s) = (sI  A)'B and G1(s) = (sI  A)IT (8.1,3)
The feedback matrix H(s) represents any analytical operations performed on the out put signal. The transfer function matrix J(s) represents any operations performed on the error signal e, as well as the servo actuators that drive the aerodynamic control surfaces, including the inertial and aerodynamic forces (hinge moments) that act on them. The servo actuators might be hydraulic jacks, electric motors, or other devices. This matrix will be a significant element of the system whenever there are power assisted controls or when the aircraft has a flybywire or flybylight AFCS.
From (8.1,l) we can derive expressions for the three main transfer function ma trices. By eliminating x, e, c, and z we get
[I + (DG + E)JH]y = (DG + E)Jr  (DG + E)JHn + DG'g (8.1,4)
from which the desired transfer functions are
G,, = [I + (DG + E)JH]'(DG + E)J (a>
G,, =  [I + (DG + E)JH] ' (DG + E)JH (b) ( 8 . 1 3
G,, = [I + (DG + E)JH]'DG' (c)
262 Chapter 8. Closed Loop Control
The matrices that appear in (8.1,5) have the following dimensions:
D(q x n); G(n x m); E(q X m); J(m X p); H(p X q); Gf(n X 2)
The forwardpath transfer function, from e to y, is
F(s) = (DG + E)J; dimensions (q x p)
so the preceding transfer functions can be rewritten as
Note that F and H are both scalars for a singleinput, singleoutput system. When the linear system model is being formulated in state space, instead of in
Laplace transforms, then one procedure that can be used (see Sec. 8.8) is to generate an augmented form of (8.1,2). In general this is done by writing time domain equa tions for J and H, adding new variables to x, and augmenting the matrices A and B accordingly. An alternative technique for using differential equations is illustrated in Sec. 8.5. There is a major advantage to formulating the system model as a set of dif ferential equations. Not only can they be used to determine transfer functions, but when they are integrated numerically it is possible, indeed frequently easy, to add a wide variety of nonlinearities. These include second degree inertia terms, dead bands and control limits (see Sec. 8.5), Coulomb friction, and nonlinear aerodynamics given as analytic functions or as lookup tables.
AN AERODYNAMICS VIEWPOINT
It is frequently helpful to view a feedback loop as simply a method of altering one of the airplane's inherent stability derivatives. When one of the main damping deriva tives, L,, M,, or N,., is too small, or when one of the two main stiffnesses Ma or N p is not of the magnitude desired, they can be synthetically altered by feedback of the ap propriate control. Specifically let x be any nondimensional state variable, and let a control surface be displaced in response to this variable according to the law
A6 = k h r ; k = const
(Here k is a simplified representation of all the sensor and control system dynamics!) Then a typical aerodynamic force or moment coefficient Ca will be incremented by
This is the same as adding a synthetic increment
to the aerodynamic derivative Cax. Thus if x be yaw rate and 6 be rudder angle, then the synthetic increment in the yawdamping derivative is
which might be the kind of change required to correct a lateral dynamics problem. This example is in fact the basis of the oftenapplied "yaw damper," a stabilityaug
8.1 General Remarks 263
mentation feature. Again, if x be the roll angle and 6 the aileron, we get the entirely new derivative
C1, = kC1, (8.1,lO)
the presence of which can profoundly change the lateral characteristics (see Exercise 8.1).
SENSORS
We have already alluded to the general nature of feedback control, and the need to provide sensors that ascertain the state of the vehicle. When human pilots are in con trol, their eyes and kinesthetic senses, aided by the standard flight information dis played by their instruments, provide this information. (In addition, of course, their brains supply the logical and computational operations needed, and their neuromus cular systems all or part of the actuation.) In the absence of human control, when the vehicle is under the command of an autopilot, the sensors must, of course, be physi cal devices. As already mentioned, some of the state information needed is measured by the standard flight instrumentsair speed, altitude, rate of climb, heading, etc. This information may or may not be of a quality and in a form suitable for incorpora tion into an automatic control system. In any event it is not generally enough. When both guidance and attitudestabilization needs are considered, the state information needed may include:
Position and velocity vectors relative to a suitable reference frame.
Vehicle attitude (8, 4, $).
Rotation rates (p, q, r).
Aerodynamic angles (a, P). Acceleration components of a reference point in the vehicle.
The above is not an exhaustive list. A wide variety of devices are in use to measure these variables, from Pitotstatic tubes to sophisticated inertialguidance platforms. Gyroscopes, accelerometers, magnetic and gyro compasses, angle of attack and sideslip vanes, and other devices all find applications as sensors. The most common form of sensor output is an electrical signal, but fluidic devices have also been used. Although in the following examples we tend to assume that the desired variable can be measured independently, linearly, and without time lag, this is of course an ideal ization that is only approached but never reached in practice. Every sensing device, together with its associated transducer and amplifier, is itself a dynamic system with characteristic frequency response, noise, nonlinearity, and crosscoupling. These at tributes cannot finally be ignored in the design of real systems, although one can use fully do so in preliminary work. As an example of crosscoupling effects, consider the sideslip sensor assumed to be available in the gust alleviation system of Sec. 8.9. Assume, as might well be the case, that it consists of a sideslip vane mounted on a boom projecting forward from the nose. Such a device would in general respond not only to /3 but also to atmospheric turbulence (side gusts), to roll and yaw rates, and to lateral acceleration a , at the vane hinge. Thus the output signal would in fact be a complicated mathematical function of several state variables, representing several
264 Chapter 8. Closed Loop Control
feedback loops. The objective in sensor design is, of course, to minimize all the un wanted extraneous effects, and to provide sufficiently high frequency response and low noise in the sensing system.
This brief discussion serves only to draw attention to the important design and analytical problems related to sensors, and to point out that their real characteristics, as opposed to their idealizations, need finally to be taken into account in design.
8.2 Stability of Closed Loop Systems
For linear invariant systems such as we have discussed above, the methods available for assessing stability include those used with open loop systems. One such method is to formulate the governing differential equations, find the characteristic equation of the system, and solve for its roots. Another is to find the transfer function from input to output and determine its poles. With the powerful computing methods available, it is feasible to plot loci of the roots (or poles) as one or more of the significant design parameters are varied, as we shall see in examples to follow. For the multivariable highorder systems that commonly occur in aerospace practise this is a very useful technique.
Let us now consider the stability of the loop associated with one particular in putloutput pair in the light of (8.1,6a). Since p = 1 and F and H are scalars, the trans fer function is
The transfer functions F(s) and H(s) are ratios of polynomials in s, that is, F(s) = N,lD,, and H(s) = N21D2. Equation (8 .2 , l ) then leads to
The characteristic equation is evidently
This should be contrasted with the characteristic equation for the airframe alone, which is D(s) = 0, where D is the denominator of G(s) . The block diagram corre sponding to (8 .2 , l ) is shown in Fig. 8.2. FH is the open loop transferfunction, that is, the ratio of feedback to error, Z/Z. Its absolute value IF^ is the open loop gain.
The stability of the system can be assessed from the frequency response F(io)H(io). It is clear that if there is a frequency and open loop gain for which FH =  1 then un
Figure 8.2 consolidated block diagram of feedback controller.
8.2 Stability of Closed Loop Systems 265
Im
1
K = 8. Unstable Re
0
 1
I I I I
2  1 0
Figure 8.3 Nyquist diagram.
der those conditions the denominator of (8.2,l) is zero and G,,(iw) is infinite. When these conditions hold, the feedback signal that is returned to the junction point is pre cisely the negative of the error signal that generated it. This means that the system can oscillate at this frequency without any input. This is exactly the situation with the whistling public address system. For then the acoustic signal that returns to the mi crophone from the loudspeakers, in response to an input pulse, is equal in strength to the originating pulse. Clearly the point ( 1,0) of the complex plane has special sig nificance. Nyquist (1932) has shown how the relationship of the frequency response curve (the Nyquist diagram) of FH to this special point indicates stability (see Fig. 8.3). In brief, if the loop gain is < 1 when the phase angle is 1 80°, or if the phase is < 180" when the gain is unity, then the system is stable. The amounts by which the curve misses the critical point define two measures of stability, the gain margin and phase margin, illustrated on the Nichols diagram (see McLean, 1990) of Fig. 8.4. The
0.1 360 300 2 4 0 1 8 0 1 20 60 0
Phase, degrees
Figure 8.4 Nichols diagram, K = 2.
266 Chapter 8. Closed Loop Control
examples of Figs. 8.3 and 8.4 are for the open loop transfer function
8.3 Phugoid Suppression: Pitch Attitude Controller
The characteristic lightly damped, lowfrequency oscillation in speed, pitch attitude, and altitude that was identified in Chap. 6, was seen in Chap. 7 to lead to large peaks in the frequencyresponse curves (Figs. 7.14 to 7.18) and long transients (Fig. 7.20). Similarly, in the controlfixed case, there are large undamped responses in this mode to disturbances such as atmospheric turbulence. These variations in speed, height, and attitude are in fact not in evidence in actual flight; the pilot (human or automatic) effectively suppresses them, maintaining flight at more or less constant speed and height. The logic by which this process of suppression takes place is not unique. In principle it can be achieved by using feedback signals derived from any one or a combination of pitch attitude 0, altitude h, speed v, and their derivatives. In practice, the availability and accuracy of the state information determines what feedback is used.
Since the phugoid oscillation cannot occur if the pitch angle 0 is not allowed to change (except when commanded to), a pitchattitudehold feature in the autopilot would be expected to suppress the phugoid. This feature is commonly present in air plane autopilots. We shall therefore look at the design of an attitude hold system for the jet transport of our previous examples. Pitch attitude is readily available from ei ther the real horizon (human pilot) or the vertical gyro (autopilot). Consider the con troller illustrated in Fig. 8.5. From (8.2,l) we see that the overall transfer function is
If we write Go,&) = N(s)lD(s), and J(s) = N'(s)lDr(s), then the characteristic equa tion is
D(s)Dr(s) + N(s)N1(s) = 0 (8.3,2)
To proceed further we need explicit expressions for the above transfer functions. Since 0 is an important variable in both the short period and phugoid modes, it might be expected that neither of the two approximate transfer functions for derived in Sec. 7.7 would serve by itself. We therefore use the exact transfer function derived
controller aircraft
1 1
Figure 8.5 Pitch attitude controller.
8.3 Phugoid Suppression: Pitch Attitude Controller 267
k , = 0
Phugoid E:l , ,J k , =  0 5
O.ll 0.75 0.50 0.25 0
Real s
Figure 8.6 Root locus of pitch controller with proportional control.
from the full system of linearized longitudinal equations of motion. Then N(s) is given by (7.7,2) and D(s) by (6.2,2). The result is
As to J(s), a reasonable general form for this application is
For obvious reasons, the three terms on the right hand side are called, respectively, integral control, proportional control and rate control, because of the way they oper ate on the error e. The particular form of the controlled system, here G,,<,(s), deter mines which of k, , k,, k, need to be nonzero, and what their magnitudes should be for good performance. Integral control has the characteristic of a memory, and steady state errors cannot persist when it is present. Rate control has the characteristic of an ticipating the future values of the error and thus generates lead in the control actua tion. In using (8.3,4), we have neglected the dynamics of the elevator servo actuator and control surface, which would typically be approximated by the firstorder trans fer function 1/(1 + rs). Since the characteristic time of the servo actuator system, r, is usually a small fraction of a second, and we are interested here in much longer times, this is a reasonable approximation.
For the example airplane at the chosen flight condition it turns out that we need all three terms of (8.3,4) to get a good control design. This might not always be the case. Let us first look at the use of proportional control only, in which case J is a con stant gain, k,. To select its magnitude, we use a root locus plot2 of the system, Fig. 8.6, in which the locus of the roots of the characteristic equation of the closed loop system are plotted for variable gain k,. We see that at a gain of about 0.5 the phugoid mode is nearly critically damped, that is, it is about to split into two real roots. At this gain, the phugoid oscillation is effectively eliminated. We note that at
'The term root locus is used throughout this chapter with the meaning ordinarily ascribed to it in the control theory literature.
268 Chapter 8. Closed b o p Control
the same gain the short period roots have moved in the direction of lower damping. The response of the aircraft to a unit step command in pitch angle with only propor tional control is shown in Fig. 8 . 7 ~ . It is clear that this is not an acceptable response. There is a large steadystate error (steadystate error is a feature of proportional con trol) and the shortperiod oscillation leads to excessive hunting. The steadystate er ror could be reduced by increasing k, (see Exercise 8.2), but this would further de crease the short period damping.
2
1.5
0 1
0.5
0 0 20 40 60 80 100 120
Time, s
(a)
Time, s
(b )
Time, s
(c )
Figure 8.7 Response of pitch angle to unit step command. (a) With proportional control. (b) With proportional plus integral control. (c ) With proportional, integral, and rate control.
8.3 Phugoid Suppression: Pitch Attitude Controller 269
We digress briefly to explore the reason for the damping behavior. It was noted previously (Sec. 6.8 and Exercise 6.4) that the term of nexttohighest degree in the characteristic equation gives "the sum of the dampings." That is, the coefficient of s3 in (8.3,3) is the sum of the real parts of the shortperiod and phugoid roots. Now when J = k2 the closed loop characteristic equation (8.3,2) becomes D(s) + k,N(s). That is we add a second degree numerator to a fourth degree denominator, leaving the coefjcient of shnchanged. Thus any increase in the phugoid damping can only come at the expense of that of the shortperiod mode. This is exactly what is seen in Fig. 8.6. The shifts of the two roots in the real direction are equal and opposite.
To eliminate the steadystate error, we use integral control and choose
The result is shown in Fig. 8.7b. The steadystate error has been eliminated, but the shortperiod oscillation is now even less damped. Now the damping of the short period mode is governed principally by M , [see (6.3,14)], so in order to improve it we should provide a synthetic increase to M,. A signal proportional to q is readily ob tained from a pitchrate gyro. Since q = 8 in the system model we are using, we ac complish this by adding a third term to J:
The result, shown in Fig. 8 . 7 ~ is an acceptable controller, with little overshoot and no steadystate error. A commanded pitch attitude change is accomplished in about 10 sec. Note that in this illustration, all the constants in (8.3,4) are the same, that is, 0.5. Finetuning of these could be used to modify the behavior to reduce the over shoot or speed up the response. Throughout this maneuver, the elevator angle remains less than its steadystate value (which it approaches asymptotically), so that the gains used are indeed much smaller than the elevator control is physically capable of pro viding (see Exercise 8.2).
The preceding analysis does not reveal the underlying physics of why k2 damps the phugoid. This can be understood as follows. An angle 8 in the low frequency phugoid implies vertical velocity (i.e., h = V8). Now a positive elevator angle pro portional to a slowly changing 8 implies a negative increment in angle of attack and hence in the lift as well. Thus k, leads to a vertical force (downward) 180" out of phase with the vertical velocity (upward), exactly what is required for damping.
Although the phugoid oscillation has been suppressed quite successfully by the strategy employed above, it should be remarked that in the example case neither the speed nor the altitude has been controlled. As a consequence, the speed drifts rather slowly back to its original value, and the altitude to a new steady state.
Finally it should be noted that a controller design that is correct for one flight condition, in this case high speed at high altitude, may not be acceptable at all speeds and altitudes, for example, landing approach. In the real world of AFCS design, this problem leads, as in most engineering design, to compromises between conflicting requirements. If the economics of the airplane justifies it, gain scheduling can be adopted; that is, the control gains are made to be functions of speed, altitude, and configuration.
270 Chapter 8. Closed Loop Control
8.4 Speed Controller
The phugoid makes its presence known not only in the form of transient perturba tions from a steady state, but also in maneuvers, as illustrated in Sec. 7.7. We saw there for example that in changing from level to climbing flight by opening the throt tle (Fig. 7.21) there results a protracted, weakly damped approach to the new state that would take more than 10 min to complete. Transitions from one value of y to an other are obviously not made in this manner, and the pilot suppresses the oscillation in this case as well. Provided that the correct 6 is known for the climb condition, the same technique as discussed above would work, that is, control operating on pitch attitude error. We illustrate an alternative concept that does not require any knowl edge of the final correct pitch attitude, but that uses speed error alone. It is not self evident how speed should be controlled, in the light of the discussion in Sec. 7.1. We saw there that both elevator and throttle influence the speed, but that the short and longterm effects of each of these controls are quite differentthe throttle principally affects the speed only in the short term. For a change of steadystate speed, the eleva tor must be used. Clearly, a sophisticated speed control might use both. We shall see in this example, however, that when the primary aim is to suppress the phugoid, which is a very long period oscillation, the goal can be achieved with the elevator alone. Figure 8.8 shows the system.
The command is the speed u, and the feedback signal is the actual speed u. For output we choose speed and flightpath angle, that is, y = [u y]T. The control vector is c = [ae 6,IT of which only the elevator is in the feedback loop. Since the con trolled variable is u, which does not change appreciably in the shortperiod mode, we can use the phugoid approximation for the aircraft transfer function matrix G(s), which is the (2 X 2) matrix of transfer functions from c to y:
Two of the elements of G are implicit in (7.7,7), since G,, = Go,  G,, where S stands for either 6, or 6,. The remaining two are (see Exercise 8.4)
x8p 4442 G,, =    m f(s>
 Figure 8.8 Speed controller.
8.4 Speed Controller 271
Ultimately we shall want to calculate the time responses of u and y to a throttle input 6,. So the transfer functions we need are the two corresponding closed loop transfer functions. If we denote these by &,,,js) and Gyg(s), respectively, we find that they are given in terms of the aircraft transfer functions by (see Exercise 8.4)
Each of the aircraft transfer functions in (8.4,3) can, as usual, be expressed as a ratio of two polynomials, for example:
Nu ,<,
=  f (s) . etc.
and
When this is done (8.4,3) becomes
We know that the denominator of a transfer function is the characteristic polynomial. We also know that a linear invariant system of the kind under discussion can have only one independent characteristic equation. Thus we have an apparent paradox, since the denominator of (8.4,6) is not the same as that of (8.4,5), having the extra factor f(s). Now it can be shown (see Exercise 8.6) that f is a factor of the bracketed term in the numerator of (8.4,6), and hence that it divides out of the right side and leaves the same characteristic polynomial as in (8.4,s).
As indicated above, the secondorder phugoid approximation should be expected to be reasonable for this case. We shall therefore use it to choose the gains in J(s), but at the end will check the solution for suitability with the exact fourthorder equations. To this end we examine the effect of J(s) on the characteristic equation, that is, on
f ( m J + N,N,,<, = 0
f (s) is given by (6.3,9):
f(s) = AS' + Bs + C
N,,, is given by (7.7,7):
and for J(s) we use
J(s) = k, + k2s
so that D, = 1 and N, = k, + k,s. Note that the k,s term implies a signal proportional to acceleration. Such a signal could be obtained from an xaxis accelerometer or by
272 Chapter 8. Closed Loop Control
I I I I 0.8 0.6 0.4 0.2 0
Real s
Figure 8.9 Speed controller. Root locus plot of GUae. Phugoid approximation.
differentiating the signal from the speed sensor. The closed loop characteristic equa tion then becomes:
A's2 + B's + C' = 0 (a) where A' = A + alk2 (b) (8.49)
B' = B + a,k, + a&2 (c)
C' = C + a&, (4 The numerical values of the constants for the example jet transport are
A = 2.721 X 10'
B = 2.633 X lo5 C = 1.376 X lo5
a , = 8.218 X 10'
a, = 3.653 X 10'
0.002 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 Proportional, k, radlfps
Figure 8.10 Speed controller gain relation for 6 = 1. Phugoid approximation.
8.4 Speed Controller 273
1 0 0 20 40 60
T~me, s
Figure 8.11 Speed controllerphugoid approximation. Speed response to throttle input.
To assess what range of values of k , and k, would be appropriate, we use three guides:
/
I. A reasonable elevator angle for, say, a 10 fps (3.048 mls) speed error
2. The root locus plot for GUse 3. The graph of k, vs. k , for critical damping
A
J = ,005 (3s + I ) Input: throttle. 6, =  1 6
(1) The first of these is arrived at by noting that l o of elevator for 10 fps speed error gives a k , of 0.0017 radlfps. (2) The root locus plot is shown on Fig. 8.9 and indicates that the open loop roots can be moved very appreciably with a proportional gain as low as 0.005. (3) For the third guide, we note that critical damping corresponds to ~ ' 2  4A1C' = 0. With the aid of (8.4,9) and (8.4,10) this leads to an algebraic rela
tion between k , and k, that is solved for the graph shown on Fig. 8.10. The useful range of gains is the space below the curve, which corresponds to damped oscilla tions. The farther from the curve, the more overshoot would be expected in the re sponse. We have for illustration arbitrarily chosen the gains indicated by the point marked on the graph, without regard for whether it is optimum. When used to calcu late the response of airplane speed to application of a negative step in thrust, with the phugoid approximation, the result is as shown in Fig. 8.1 1. The throttle input corre
Figure 8.12 Speed controllerexact equations. Speed response.
274 Chapter 8. Closed Loop Control
0.025 J = 0.005 (3s + 1)
U 6 =' m P 6 / %s
2 0.05
$ w
0.1 I I I I I I J 0 20 40 60 80 100 1
Time, s
Figure 8.13 Speed controller~xact equations. Gamma response.
sponds to a steadystate descent angle of a little less than 3". The maximum speed er ror, which is seen to be less than 3 fps at an initial speed of 774 fps, would probably not be perceptible to the pilot. This suggests that the chosen gains are probably not too small. The maximum elevator angle during this maneuver is less than 2" (see Fig. 8.14) so the gains are not excessive either.
To assess the performance of the controller with certainty, it is necessary to use the exact equations. The full matrix A for this example is (6.2,1), and B is (7.6,4). The most important elements of the solution are displayed in Figs. 8.12 to 8.14. The result for the speed in Fig. 8.12 confirms that the phugoid approximation is indeed good enough for preliminary design. Figure 8.13 demonstrates that the steadystate flight path angle is reached, with a small overshoot, in about 20 s. Figure 8.14 demonstrates that the elevator angle required to achieve this is small. To understand the physics of the maneuver, it is helpful to look at the angle of attack variation, graphed in Fig. 8.15. It shows that there is a negative "pulse" in a that lasts about 10 s. This causes a corresponding negative pulse in lift, which is the force perpendic ular to the flight path that is required to change its direction.
Finally, these graphs should be contrasted with those of Fig. 7.21, which show the uncontrolled response to throttle. Feedback control has made a truly dramatic dif ference!
0.05 0 10 20 30
Time, s
Figure 8.14 Speed controllerexact equations. Elevator angle.
8.5 Altitude and Glide Path Control 275
Figure 8.15 Speed controllerexact equations. Angle of attack response
8.5 Altitude and Glide Path Control
One of the most important problems in the control of flight path is that of following a prescribed line in space, as defined for example by a radio beacon, or when the air plane flies down the ILS glide slope. We discuss this case by considering first a sim ple approximate model that reveals the main features, and then examining a more re alistic, and hence more complicated case.
FLIGHT AT EXACTLY CONSTANT HEIGHTSPEED STABILITY
The first mathematical model we consider can be regarded as that corresponding to horizontal flight when a "perfect" autopilot controls the angle of attack in such a way as to keep the height error exactly zero. The result will show that the speed variation is stable at high speeds, but unstable at speeds below a critical value near the mini mum drag speed. Neumark (1950) recounts that this criterion was first discovered in 1910 by PainlevC, and that it was at first accepted by aeronautical engineers and sci entists, but later, on the basis of the theory of the phugoid, which showed no such ef fect, was rejected as false. In fact, to the extent that pilots can control height error by elevator control alone, that is, to the extent that they approximate the ideal autopilot we have postulated, the instability at low speed will be experienced in manual flight. Since speed variation is the most noticeable feature of this phenomenon, it is com monly referred to as speed stability.
We could analyze this case by applying (4.9,18) to the stated flight condition. However, it is both simpler and more illuminating to proceed directly from first prin ciples. The airplane is flying on a horizontal straight line at variable speed V. It is im plied that CY is made to vary, by controlling S,, in such a way that the lift is kept ex actly equal to the weight at all times. The equation of motion is clearly
where T is the horizontal component of the thrust, and D is the drag. Since the speed cannot change very rapidly, then neither does a, and we can safely ignore any effects of q and iu on lift and drag. Consequently, T and D are simply the thrust and drag or
276 Chapter 8. Closed Loop Control
D ( L = W I
v* Figure 8.16 Performance graph.
dinarily used in performance analysis, as displayed in Fig. 8.16. We denote the refer ence thrust and drag by To and Do and define the stability derivatives
T,= aTlav and Dv= a ~ l a v
so that T  D = (To + TvAV)  (Do + DvAV)
Since V = Vo + AV, and To = Do, (8.51) becomes
This firstorder differential equation has the solution
with
T , and Dv are the slopes of the tangents to the thrust and drag curves at their intersec tion. If they intersect at a point such as P in Fig. 8.16, then T , < D , A is negative, and the motion is stable. If, on the other hand, the flight condition is at a point such as Q, the reverse is the case. A is then >0, and the motion is unstable. If when flying at point Q there is an initial error in the speed, then it will either increase until it reaches the stable point P or it will decrease until the airplane stalls. The stable and unstable regimes are bounded by the speed V*, which is where the thrust curve is tan gent to the drag curve. V* will be the same as V,, of Fig. 7.1 if T, = 0. Hence the appellation "back side of the polar" is used to describe the range V < V*, with refer ence to the portion of the aircraft polar (the graph of CL vs. C,) for which CL is greater than that for maximum WD.
Although we have analyzed only the case of horizontal flight, the result is similar for other straightline flight paths, climbing or descending (see Exercise 8.8). Flight in the unstable regime can indeed occur when an airplane is in a low speed climb or landing approach. This speed instability is therefore not entirely academic, but can present a real operational problem, depending on by what means and how tightly the aircraft is constrained to follow the prescribed flight path. An important point insofar as AFCS design is concerned is that for speeds less than V* it is not possible to lock
8.5 Altitude and Glide Path Control 277
exactly onto a straightline flight path, and at the same time provide stability, using the elevator control alone, no matter how sophisticated the controller! To achieve sta bility it is mandatory to use a second control. This would most commonly be the throttle, but in principle spoilers that control the drag could also be used.
EXAMPLEAN ALTITUDE CONTROLLER
In view of the above, we illustrate an altitude controller that also incorporates control of speed, using once again our example jet airplane. This time we make the system model more realistic by including firstorder lag elements for the two controls: that for the elevator is mainly associated with its servo actuator (time constant 0.1 s); and that for the throttle with the relatively long time lag inherent in the build up of thrust of a jet engine following a sudden movement of the throttle (time constant 3.5 s). An other feature that is incorporated to add realism to the example is a thrust limiter. Be cause transport aircraft inherently respond slowly to changes in thrust, the gains cho sen to give satisfactory response for very small perturbations in speed will lead to a demand for thrust outside the engine envelope for larger speed errors. We have there fore included a nonlinear feature that limits the thrust to the range 0 5 T 5 l.lTo. This contains the implicit assumptions (quite arbitrary) that the airplane, flying near its ceiling, has 10% additional thrust available, and that idling engines correspond to zero thrust.
At the same time this example illustrates an alternative approach to generating the analytical model of the system, in terms of its differential equations. In the previ ous illustrations we have, by contrast, used what may be termed "transfer function al gebra" to arrive at transfer functions of interest, and then used these to obtain what ever results were desired. The end result of the modeling to follow is a system of differential equations that is then integrated to get time solutions. Since the limiter is inherently a nonlinear element, it is in any case not possible to include it in a transfer function based analysis.
The system block diagram is shown in Fig. 8.17. The commanded speed and alti tude are the reference values u, and h,,, so that the two corresponding error signals are Au and Ah. Note that h is the negative of z, used in Chap. 4. The inner loop for 8 is that previously studied in Sec. 8.3, with the J,(s) modified to account for the
Figure 8.17 Altitudehold controller.
278 Chapter 8. Closed Loop Control
elevator servo actuator. The logic of the outer loop that controls h warrants explana tion. If there is an initial error in h, say the altitude is too low, then in order to correct it, the airplane's flight path must be deflected upward. This requires an increase in angle of attack to produce an increase in lift. The angle of attack and the resulting lift could of course be produced by using an angle of attack vane as sensor, and no doubt an angle of attack commanded to be a function of height error would be very effec tive. It might be preferred, however, to use the vertical gyro as the source of the sig nal, and since shortterm changes in 0 are effectively changes in a, then much the same result is obtained by using 0 as the commanded variable. We have chosen to use stability axes, so that in the steady state, when Ah is zero, the correct value of 0 is also zero. Thus, in summary, the system commands a pitch angle that is proportional to height error and the inner loop uses the elevator to make the pitch angle follow the command. While all this is going on the speed will be changing because of both grav ity and drag changes. The quickest and most straightforward way of controlling the speed is with the throttle, and the third loop accomplishes that. (The symbols y, and y, denote the inputs to the limiter and the airframe, and are elements of the state vec tor derived below.)
The Differential Equations
The basic matrix differential equation of the airframe, with 0, = 0, and AzE = Ah is obtained from (4.9,18) and (7.6,4). Up to this point, we have neglected engine dy namics and in effect regarded thrust as proportional to 6,. The matrix B of (7.6,4) is structured in that way. To accommodate the facts that 6, actually represents the throt tle setting, not the thrust, and that the two are dynamically connected, we need to in troduce two new symbols, y, and c*. The quantity y,, when multiplied by X,p, etc. yields the aerodynamic force and moment increments AX,, etc., and c* is defined be low. The differential equation of the airframe is then
t = Ax + Bc* (8.5,4)
where x = [Au w q 0 AhIT
C* = [a, y5IT
A = [a,]
B = [b,I
To obtain the differential equations of the three control elements, we begin with their transfer functions, which are specified for this example to be
Je(s) = (aos' + a , + a2s)(l + res)'
= (a, + a,s + a2s2)(s + res2)' ( 8 . W )
C,,(s) = (b,,,' + b, + b2s)
= (bo + b,s + b2s2)ls (8.5,6)
J,(s) = 1 4 1 + 7,s) (8.5,7)
The first two of these contain proportional + rate + integral controls, all of which were found to be needed for good performance. The time domain equations that cor respond to the elements of the controller are then as follows (verify this by taking their Laplace transforms):
8.5 Altitude and Glide Path Control 279
After substituting the expressions for the two error signals, (8.5,8) yield three equa tions for the controls
T,A$, + A& = ka2Ah  a2B  ka,Ah  a,8  ka,Ah  a,0 (a )
y4 = b2Aii  b,Aa  b,Au (6) (8.59)
T , ~ S = Aq,  Y4 ( c )
For convenient integration we want a system of firstorder equations and therefore have to do something about the second derivatives in (8.5,9). Since 6 = q we can re place 8 with g. For the other second derivatives, we introduce three new variables, as follows:
y , = Au (a)
y2 = Ah (b) (8.5,10)
Y3 = As, ( c )
With these definitions, (8.5,9a and b) can now be rewritten in terms of first deriva tives as
The state vector now consists of the original five variables from (8.5,4) plus the two control variables AS, and AS,, plus the five yi defined above, making 12 in all. We therefore require 12 independent equations. From the foregoing equations (8.5,4) (8.5,10), (8.5,l I) , and (8.5,9c) we can get 1 1 of the required differential equations. That for y , is obtained from (8.5,lOa) by differentiating the first component of (8.5,4) and that for y, by differentiating the fifth. The result of that operation is
Finally, the 1 1 independent differential equations are assembled as follows:
280 Chapter 8. Closed Loop Control
Solutions The above equations contain first derivatives on the right side as well as on the left side and hence are not in the canonical form. This is no impediment to numerical in tegration, however, since if the derivatives are calculated in the sequence given, each one that appears on the right has already been calculated in one of the preceding equations by the time it is needed. The twelfth and final relation needed is that which describes the limiter, in the form
From the values of C,, and C,, given in Sec. 6.2, we find that Do = To =
0.0657 W. In Sec. 7.6 it was given that 6, = 1 corresponds to a thrust of 0.3 W. It fol lows that zero thrust corresponds to y, = 0.06571.3 = 0.219. The nonlinear rela tionship for y, is therefore implemented in the computing program by a program fragment equivalent to
AS, = Y4
IF y, < 0.219 THEN A6, = 0.219 (8.5,14)
IFy4 > 0.10 THEN AS, = 0.10
where the maximum engine thrust has been assumed to be 10% greater than cruise thrust. Equations (8.5,13 and 8.5,14) are convenient for numerical integration. We have calculated a solution using simple Euler integration of the equations for the ex ample jet transport with the matrices A and B given in Secs. 6.2 and 7.6, and with the following control parameters: re = 0.1; r, = 3.5; k = 0.0002; a, = a , = a, = 0.5; bo = 0.005; b, = 0.08; b, = 0.16. Figure 8.18 shows the performance obtained in re sponse to an initial height error of 500 ft.
It is seen that the height error is reduced to negligible proportions quickly, in about 20 s, accompanied by a theta pulse of similar duration and peak magnitude about 7". Even with extreme throttle action, the speed takes more than 2 min to re cover its reference value. This length of time is inherent in the physics of the situa tion and cannot be shortened significantly by changes in the controller design. On the other hand, there is no operational requirement for more rapid speed adjustment when cruising at 40,000 ft.
The peak elevator angle needed is less than 3", but the thrust drops quickly to zero, stays there for about 30 s, then increases rapidly to its maximum. Toward the end of the maneuver the throttle behaves linearly and reduces the speed error smoothly to zero.
8.6 Lateral Control
There are five lateral state variables that can be used readily as a source of feedback signals{v, p, r, 4, $1; u from a sideslip vane or other form of aerodynamic sensor, p and r from rate gyros, and 4, $ from vertical and directional gyros. Lateral acceler ation is also available from an accelerometer. These signals can be used to drive the two lateral controls, aileron and rudder. Thus there is a possibility of many feedback loops. The implementation of some of these can be viewed simply as synthetic modi fication of the inherent stability derivatives. For example, p fed back to aileron modi
8.6 Lateral Control
600 I 1
. < 2000. rad
300 I I I I I I I I I 0 20 40 60 80 100 120 140 160 180 200
400 I I I 0 50 100 150 200
T~me, s
( b )
Figure 8.18 Altitudehold controller. (a ) Height, speed, and pitch angle. (b) Elevator and throttle controls.
fies L, (roll damper), r to the rudder modifies N , (yaw damper), and v to rudder mod ifies the yaw stiffness Nu, and so on. It is a helpful and instructive preliminary to a detailed study of particular lateral control objectives to survey some of these possible control loops. We could do this analytically by examining the approximate transfer functions given in Chap. 7. However, we prefer here to do this by way of example, using the now familiar jet transport, and using the full system model. We treat each loop as in Fig. 8.19, as a negative feedback with a perfect sensor and a perfect actua tor, so that the loop is characterized by the simple constant gain K. For each case we present a root locus plot with the gain as parameter (Fig. 8.20) (All the root loci are symmetrical about the real axis; for some, only the upper half is shown). As is con
282 Chapter 8. Closed Loop Control
I I Figure 819 Representative loop.
ventional, the crosses designate the open loop roots (poles) and the circles the open loop zeros. The pair of complex roots corresponds to the Dutch Roll oscillation; the real root near the origin is for the spiral mode; and the real root farther to the left is that of the heavily damped roll mode.
Since the root loci always proceed from the poles to the zeroes as 14 increases, the locations of the zeros can be just as important in fixing the character of the loci as the locations of the poles. The numbers on the loci are the values of the gain. Zero gain of course corresponds to the original open loop roots. The objective of control is to influence the dynamics, and the degree of this influence is manifested by the amount of movement the roots show for small changes in the gain. We have not in cluded root loci for acceleration feedback, and of the remaining ten, two show very small effects, and are therefore not included either. These two are the aileron feed backs: u += 6, and r + 6,. Each of the other eight is discussed individually below.
+ = 6 It was pointed out in Chaps. 2 and 3 that airplanes have inherent aerodynamic rotational stiffness in pitch and yaw, but that there is no such stiffness for rotations about the velocity vector. This funda mental feature of aerodynamics is responsible for the fact that air planes have to sideslip in order to level the wings after an initial roll upset. This lack can be remedied by adding the synthetic derivative
We might expect that making such a major change as adding a new aerodynamic rotational stiffness would have profound effects on the airplane's lateral dynamics. Figure 8.20a shows that this is indeed the case. The time constants of the two nonperiodic modes are seen to change very rapidly as the gain is increased, until with even a small gain, 14 < 1, that is, less than l o of aileron for l o of bank, these two modes have disappeared, to be replaced by a low fre quency, heavily damped oscillation. The Dutch Roll remains virtu ally unaffected by the aileron feedback for any modest gain.
p+= 6, This root locus is shown in Fig. 8.20b. The largest effect is on the roll mode, as might be expected, where a positive gain of unity (cor responding to a decrease in JL,~) results in a substantial reduction in the magnitude of the large real root. This is accompanied by an in crease in the spiral stability and a slight reduction in the Dutch Roll damping. A negative gain, (an increase in I L , ~ ) increases the Dutch Roll damping, shortens the roll mode time constant and causes a slight reduction in the magnitude of the spiral root (the latter not visible in the figure).
+ += 6, Because + is the integral of r, the transfer functions for + have an s factor in the denominator, and hence a pole at the origin. This is
8.6 Lateral Control 283
0.2 0.6 0.5 0 4 0.3 0.2 0.1 0 0.1
Real s
Real s
(b ) p + 6,
Figure 8.20 Root loci. (u) 4 + 6,'. (b) p + 6,. (c) $ + 6,. (4 u + 6,. (e) p + 6,. ( f ) r + 6,. (g) 4 + 6,. ( h ) $+ S,.. (i) STOL airplane; r + 6,.
seen in Fig. 8 . 2 0 ~ . The expansion theorem (A.2,10) shows that the zero root of the characteristic equation leads to a constant in the so lution for +. This is consistent with the fact that the reference direc tion for + is arbitrary.
The feedback of I) to aileron has little relative effect on the Dutch Roll and rolling modes. Its main influence is seen on the spi ral and zero roots, which are quite sensitive to this feedback. For negative gain (stick left for yaw to the right) these two modes rapidly combine into an oscillatory mode that goes unstable by the time the gain is 0.5 (0.5" aileron for l o yaw). For all positive gains, there is an unstable divergence.
v + 6, This feedback (Fig. 8.20d) represents rudder angle proportional to sideslip, with positive gain corresponding to an increase in Nu. Note
Dutch roll
Spiral




i
 1
0 K = 0.5 
0  Roll
I I I
i
0.5 0  ., 
I I
284 Chapter 8. Closed Loop Control
0.2 0.5 0.03 0.01 0.01 0.01 0.03 0.05
Real s
(c ) W +
0.2 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1
Real s
Figure 8.20 (Continued )
that a gain of 0.001 for v corresponds to 6,lP = 0.774. The princi pal effect is to increase the frequency of the Dutch Roll while simul taneously decreasing the spiral stability, which rapidly goes unstable as the gain is increased. The reverse is true for negative gain. The roll mode remains essentially unaffected.
p , 6, Roll rate fed back to the rudder has a large effect on all three modes. For positive gain (right rudder for roll to the right) the damping of the Dutch Roll is increased quite dramaticallyit is quadrupled for a gain of about 0.2" rudderldegls of roll rate. This is counterintuitive (see Exercise 8.10). At the same time, the damping of the roll mode is very much diminished, and that of the spiral mode is increased. With further increase in gain the two nonperiodic modes combine to form an oscillation, which can go unstable at a gain of about 0.4.
r 6 The large effects shown in Fig. 8.20f for the yaw damper case are what would be expected. As an aid in assessing the damping perfor
8.6 Lateral Control 285
 
0.6 0.5 0.4 0 3 0.2 0.1 0 0.1 Real s
(e) p t 6,
1.2
1.2 1.5 1 .O 0 5 0 0.5 1 .O
Real s
Figure 8.20 (Continued)
mance, two lines of constant relative damping 5 are shown on this figure. Negative gain corresponds to left rudder when yawing nose right. A very large increase in Dutch Roll damping is attained with a gain of  1, at which point there is a commensurate gain in the spi ral damping. There is some loss in damping of the roll mode. The beneficial effects of yawrate feedback are clearly evident from this figure. The behavior for larger negative gains, beyond about  1.4, is especially interesting. For this airplane at this flight condition, the two real roots combine to form a new oscillation, the damping of which rapidly deteriorates with further increase in negative gain. This feature complicates the choice of gain for the yaw damper. The pattern shown is not the only one possible. Figure 8.20i is the corre sponding root locus for the STOL airplane of Sec. 8.9, flying at 10,000 ft and 200 k. It illustrates the importance of the location of the zeroes of the closed loop transfer function. For the jet transport the real zero, z is to the left of the real roll mode root p. For the
286 Chapter 8. Closed Loop Control
0.9  0 8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1
Real s
k) $ + 6,
Real s
(h) g t 6,
0 03 0.02 SF Real  2 v


6 6   ...  /"\
 P, pole Z, zero


I I I I
(i) STOL Airplane; r + 6,
Figure 8.20 (Continued)
8.7 Yaw Damper 287
STOL airplane the reverse is the case. The direction of the locus emanating from this root is therefore opposite in the two cases, with a consequent basic difference in the qualitative nature of the dynam ics. For the STOL airplane the Dutch Roll root splits into a real pair, one of which then combines with the spiral root to form a new low frequency oscillation. In viewing Fig. 8.20i it should be noted that it was drawn for the nondimensional system model, and hence the nu merical values for the roots and the gains are not directly compara ble with those of Fig. 8.20f.
6 Feeding back bank angle to the rudder produces mixed results (Fig. 8.20g). When the gain is negative, the spiral mode is rapidly driven unstable. On the other hand if it is positive, to improve the spiral, the Dutch Roll is adversely affected.
$+ 6, The consequence of using heading to control the rudder is also equivocal. If the gain is positive (heading right induces right rudder) the null mode becomes divergent. If the gain is negative the two nonperiodic modes form a new oscillation at quite small gain that quickly becomes unstable.
8.7 Yaw Damper
Yaw dampers are widely used as components of stability augmentation systems (SAS); we saw the potential beneficial effects in Fig. 8.20f At first glance the yaw damper would appear to be a very simple application of feedback control princi plesjust use Fig. 8.20f as a guide, select a reasonable gain, and add a model for the servo actuator/control dynamics. However, it is not really that simple. There is an other important factor that has to be taken into accountnamely that during a steady turn, the value of r is not zero. If, in that situation, the yaw damper commands a rud der angle because it senses an r, the angle would no doubt not be the right one needed for a coordinated turn. In fact during a right turn the yaw damper would al ways produce left rudder, whereas right rudder would usually be required (see Fig. 7.24). This characteristic of the yaw damper is therefore undesirable. To eliminate it, the usual method is to introduce a highpass or "washout" filter, which has zero gain in the steady state and unity gain at high frequency. The zero steady state gain elimi nates the feedback altogether in a steady turn. The system that results is pictured in Fig. 8.21, where the meaning of the filter time constant is illustrated. For the servo actuatorlrudder combination of this large airplane we assume a first order system of time constant 0.3 sec.
The closed loop transfer function for Fig. 8.21 is readily found to be [see (8.2,1)]
This transfer function was used to calculate a number of transient responses to illus trate the effects of J(s) and W(s).
As a reference starting point, Fig. 8.22 shows the open loop response (W = 0) to
288 Chapter 8. Closed Loop Control
Rudder and a~rframe servo actuator
3.333K filter J =  W = S
s + 3.333 s + a
Two = 1  a
Figure 8.21 Yaw damper.
a unit impulse of yaw rate command r,. It is evident that there is a poorly damped os cillatory response (the Dutch Roll) that continues for about 2 min and is followed by a slow drift back to zero (the spiral mode). Fig. 8.22a shows that the control dynam ics (i.e., J(s)) has not had much effect on the response.
Figure 8.23 shows what happens when the yaw damper is turned on with the same input as in Fig. 8.22. It is seen that the response is very well damped with either of the two gains shown, which span the useful range suggested by Fig. 8.20f, and that the spiral mode effect has also been suppressed.
It remains to choose a time constant for the washout filter. If it is too long, the washout effect will be insufficient; if too short, it may impair the damping perfor mance. To assist in making the choice, it is helpful to see how the parameter a =
l/rwO affects the lateral roots. Figure 8.24 shows the result for a gain of K =  1.6. The roots in this case consist of those shown in Fig. 8.20f plus an additional small real root associated with the filter. It is seen that good damping can be realized for values of a up to about 0.3, that is, for time constant r down to about 3 s. This result is very dependent on the gain that is chosen. While the oscillatory modes are behav ing as displayed, the real roots are also changingthe roll root decreasing in magni tude from 2.31 at a = 0 to  1.95 at a = 0.32. The new small real root starts at the origin when a = 0 and moves slowly to the left, growing to 0.00464 at a = 0.32. When the filter time constant is 5 s the small root is 0.0038, corresponding to an aperiodic mode with t,,,, = 182 s. It is instructive to compare the performance of the yaw damper with and without the filter for an otherwise identical case. This is done in Fig. 8.25. It is seen that the main difference between them comes from the small real root, which after 5 min has reduced the yaw rate to about 5% of its peak value. This slow decay is unlikely to present a problem since the airplane heading is in evitably controlled, either by a human or automatic pilot. In either case, the residual r would rapidly be eliminated (see Sec. 8.8).
8.7 Yaw Damper 289
T~me, s (at lnit~al response. 060 s
100 200 300 400
T~me, s
(b) Longterm response
Figure 8.22 Yaw rate impulse responseopen loop, W = 0. (a) Initial response, 060 sec. (b) Longterm response.
8.8 Roll Controller
This example is of another common component of an AFCS, a control loop that maintains the wings level when flying on autopilot, or that can be commanded to roll the airplane into a turn and hold it there. We shall see in this particular case that the resulting turn is virtually truly banked, even though no special provision has been made to control sideslip.
The block diagram of the system is shown in Fig. 8.26. It incorporates the yaw damper described in the previous section and adds two additional loops. The outer loop commands 4. The 4 error is converted to a roll rate command by J,, and it is the roll rate error that is then used to drive the aileron servo actuator. If the roll rate fol lowed the command instantaneously, without lag, the bank angle response would be exponential (i.e., C$ = 4). In reality of course this ideal behavior is not achieved be cause of the airframe and servo dynamics.
290 Chapter 8. Closed Loop Control
Figure 8.23 Effect of yaw damper.
For this example we use the state vector approach to system modeling in order to provide another illustration, one that differs in detail from that of Sec. 8.5.
As usual the starting point is the basic aircraft matrix equation,
x = Ax + Bc (8.8,l)
in which x = [v p r 4IT and c = [a, &IT. The differential equations that correspond to the various control transfer func
tions in the figure are found as follows. For the yaw damper components, we have the same form of transfer functions as previously, that is,
and
From (8.8,4) we get the differential equation
For J,, we use the constant K,, and for J, we use a first order servoactuator
The relation between Sa, p, and C$c is seen from the diagram to be
8.8 Roll Controller 291
Gain K = 1.6 4 Lo
0.9 0.8 0.7 0.6 5.0 0.4 0.3 02 0.1 Re
Figure 8.24 System poles for varying washout time constant.
Tme, s (a) Short time, 030 s
T~me, s
(b) Long tlme
Figure 8.25 Effect of washout filter on yaw damper performance. (a) Short time, 030 sec. (b) Long time.
292 Chapter 8. Closed Loop Control
Figure 8.26 Roll control system.
When we substitute for J, and J, the differential equation that results is
Equations (8.8,5) and (8.8,8) are the additional equations required to augment the ba sic system (8.8,l) to accommodate the addition of the two control angles as depen dent variables. However, a little more manipulation is needed of (8.8,5). To put it in firstorder form, we define the new variable
and to put it in canonical form, we must eliminate i. This we do by using the third component equation in (8.8,l). When these steps have been taken the system can be assembled into the matrix equation
where
The matrices P and Q are:
Equation (8.8,10) was solved by numerical integration for two cases, with the results shown on Figs. 8.27 and 8.28. The various gains and time constants used were se
8.8 Roll Controller 293
Time, s
(a) $, p . and 6,
Figure 8.27 Response of roll controller to initial $of 0.262 rad (15"). (a ) 4, p, and 6,. (b) P, r, $,
6,.
lected somewhat arbitrarily, as follows:
K,,= 1.5 K O = 1.0 K , = 1.6;rU=.15 rr=.30 rw0=4.0
On the first of these figures, response to an initial bank error, we see that all the state variables experience a reasonably well damped oscillatory decay, and that the maxi
294 Chapter 8. Closed Loop Control
I 1 I I I 0 5 10 15 20 2 5 30
Time, s
, 0 5 10 15 20 2 5 30
Time, s
(b) p. r.v. 6,
Figure 8.28 Response of roll controller to roll command of 0.262 rad (15"). (a ) 4, p, 6,. (b) P, r,
$5 6,.
mum control angles required are not excessiveabout 20" for the aileron and less than 1" for the rudder. The time taken for the motion to subside to negligible levels is equal to about two Dutch Roll periods. All the variables except cC, subside to zero, whereas cC, asymptotes to a new steady state. When level flight is reestablished, the airplane has changed its heading by about 1 .go.
The second figure shows the response to a 15" bank command. The new steady state is approached with a damped oscillation that takes about 15 s to decay. The steady state is clearly a turn to the right, in which r has a constant value and cC, is in
8.9 Gust Alleviation 295
creasingly linearly. All the other variables, including the two control angles, are very small. It is especially interesting that the sideslip angle is almost zero. Clearly this controller has the capability to provide the bank angle needed for a coordinated turn. (The angle of attack and lift would of course have to be increased.)
8.9 Gust Alleviation
For the final example in this chapter, we turn to a study of the application of auto matic controls to reduce the response of an airplane to atmospheric turbulence (Byrne, 1983). This is obviously a useful goal for many flight situations, the benefits including increased passenger comfort, reduction of pilot workload, and possibly re ductions in structural loading and fatigue, and in fuel consumption. The case reported here is for a STOL airplane, which is especially vulnerable to turbulence, since the relatively low operating speed makes it more responsive to turbulence, and because its duty cycle requires it to spend relatively more time at low altitudes where turbu lence is more intense.
The numerical data used in the study were supplied by the de Havilland Aircraft Co., and although it does not apply to any particular airplane, it is representative of the class.
In this situation, where random turbulence produces random forces and moments on the airplane, which in turn result in random motion, the methods of analysis we have used in the foregoing examples, being essentially deterministic, are not applica ble. Random processes have to be described by statistical functions. Let f(t) represent such a random function. Two of the key statistical properties that characterize it are the spectrum function, derived from a Fourier analysis of f(t) and the closely related correlation function (Etkin, 1972). The spectrum function or spectral density, as it is frequently called, is denoted aff(w). The area under the Qff(w) curve that is con tained between the two frequencies w, and w, is equal to the contribution to $p (where is the meansquaredvalue of f)3 that comes from all the frequencies in the band w, 3 w2 that are contained in the Fourier representation of f .
A BASIC THEOREM
When a system with transfer function G(s) is subjected to an input with spectrum function aii(w) the spectral density @,(w) of the response r(t) is given by
where G(iw) is the frequency response function defined in Sec. 7.5. There is a gener alization of (8.9,l) available for multiple inputs (Etkin, 1972, p. 94). Figure 8.29 shows the relationships expressed in (8.9,l) for a secondorder system of moderate damping.
In the case at hand the motion studied is the lateral motion, and the forces needed are Y, L, and N. The gust vector g (Fig. 8.1) that is the source of these forces has ele ments that represent aspects of the motion of the atmosphere. It has four components
'The factor $ comes from the use of twosided spectra.
296 Chapter 8. Closed Loop Control
4
0 0.01 0.1 1 .o 10 100
61 (radls)
Figure 8.29 Response to random input.
v, is the ycomponent of the turbulent velocity, p, is the lateral gradient of the zcom ponent, p, = aw,/ay, and r,, = duglay, r,, = av,/ax. Each of these inputs is capa ble of producing aerodynamic actions on the airframe. u, acts just like v in producing forces and moments like Y,v and L,v; p, acts like p and produces moments like L,p and N g ; and r, , and r,, also produce forces and moments as a result of their effects on the relative wind at the wing and tail. The details of this theory can be found in Etkin (1981). The incremental force and moments can be expressed in terms of the gust components by the equation
Where F is a (3x4) matrix of "gust derivatives." The basic differential equation of the system is then (8.8,l) with an added term to account for the turbulence, that is,
In order to alleviate the response of the airplane, it is necessary to be specific and choose an output that is to be minimized, and then to find what control actions will
8.9 Gust Alleviation 297
u Figure 8.30 Gust alleviation system.
be successful in doing so. In the cited study, the output chosen to be minimized was a passenger comfort index (see below).
A block diagram of the system considered is given in Fig. 8.30. The state vector is the set x = [v p r 4IT and the control vector is c = [aa 8,lT. The model includes full state feedback via the (2x4) gain matrix K,, control servo actuators described by the (2x2) matrix J, and also includes the possibility of using measurements of the turbulent motion to influence the controls via the (2x4) gain matrix Kg.
We now proceed to complete the differential equation of the system. We start with the servo actuator transfer function J(s), which is given, as in our previous ex amples, by firstorder elements:
Equation (8.9,3) corresponds to the pair of differential equations
8, = (e,  Sa)/7,
8, = (e,  8r ) /~ r
or c = PC + Pe
where
We see from Fig. 8.30 that e = K,g + K 3 , so that (8.9,5) becomes
c = P K 3  PC + PK,g
298 Chapter 8. Closed Loop Control
We can now combine (8.9,2) and (8.9,7) into the augmented differential equation of the system:
This can be written more compactly, with obvious meanings of the symbols, as
z = A z + T g (8.9,9)
In the cited study, various control strategies were examined, differentiated primarily by whether or not gust "feedforward" was included (i.e., Kg # 0). When only state feedback was employed, (Kg = 0) linear optimal control theory was used to ascertain the optimum values of the gains in K,. To this end, a function has to be chosen to be minimized. The choice made was a passenger comfort index made up of a linear combination of sideways seat acceleration along with angular accelerations p and P. The seat acceleration depends on how far the seat is from the CG, so an average was used for this quantity. The optimum that resulted entailed the feedback of each of the four state variables to each of the two controls, a very complicated control system! However, it was found that there was very little difference in performance between
Frequency, hz
Figure 8.31 Lateral acceleration spectra. Rearmost seat, with yaw damper.
8.9 Gust Alleviation 299
this optimum and a simple yaw damper. The result is shown on Fig. 8.31, in the form of the spectral density of sideways acceleration of the rearmost seat. This form of plot, f@( f ) vs. log f , is commonly used. The area under any portion of this curve is also equal to the meansquare contribution of that frequency band, just as with @(f) vs. f . Results are shown for three casesthe basic airframe with fixed controls, a conventional autopilot, and the selected yaw damper. Very substantial reduction of re sponse to turbulence has clearly been achieved with a relatively simple control strat
egy. An alternative to conventional linear optimal control theory was found to be bet
ter for the case when gust measurement is assumed to be possible. It stems from a theorem of Rynaski et al. (1979). It is seen from (8.9,2) that if one could make Bc + Tg = 0 then one would have completely canceled the gust input with control action, and the airplane would fly as if it were in still air! This equality presupposes that the control is given by
that is, that B has an inverse. This would require B to be a square matrix [i.e., to be (4X4)], which in turn would require that the airplane have two more independent
Frequency, hz
Figure 8.32 Lateral acceleration spectra. Rearmost seat, with yaw damper and gust feedforward.
300 Chapter 8. Closed Loop Control
(and sufficiently powerful) controls than it actually has. Although this is not beyond the realm of imagination, it was not a feasible option in the present study. However, there is available the "generalized inverse," which provides in a certain sense the best approximation to the desired control law. The generalized inverse of B is the inverse of the (2x2) matrix BTB. This leads to the control law
(The second BT is needed to yield a (2X 1) matrix on the righthand side). This law still requires, however, that all four components of g be sensed in order to compute c. Sensing all components of g is not impossible, indeed it may not even be impractical. However, a good result can be obtained with a subset of g consisting only of v, and r,,, both of which can be measured with an aerodynamic yawmeter, a sideslip vane or other form of sensor. The end result of combining gust sensing in this way with the yaw damper, with the gust sensor placed an optimum distance forward of the CG, is shown in Fig. 8.32. It is clear that this control strategy has been successful in achiev ing a very large reduction in seat acceleration.
8.10 Exercises
8.1 Assume that an aerodynamic derivative L+ has been added to the lateral force system. What changes does this entail in the lateral characteristic equation? What implica tions do these changes have for lateral dynamics?
8.2 (a) What is the steady state 6 that results from a steady A S , = 5" for the jet transport of Sec. 8.3?
(b) For the closedloop response to a unit step input in Sec. 8.3, with J = k,, derive an expression for the steadystate error e,, as a function of k,. (Hint: start with (8.3,l)).
(c) Calculate the value of k, needed to keep e,, < 0.1" for 8, = 5".
(d) For the value of k, found in (c) what is the elevator angle at t = Of when 6, is a step input of 5"? Comment on the practicality of using k, alone to reduce e,,.
8.3 (a) With respect to Fig. 8.5, write out the transfer function for the elevator angle re sponse to 6, input.
(b) Calculate the steadystate response for the case of Fig. 8 . 7 ~ .
8.4 (a) Derive the expressions for the transfer functions G,$ and G,,p given in (8.4,2).
(b) Derive the expressions for the closed loop transfer functions given in (8.4,3).
8.5 The system of Fig. 8.8 is to be represented by the block diagram of Fig. 8.1 with x =
[u w q 6IT and c = [ S , SPIT. Write out the matrices D, E, and H. What are the di mensions of J?
8.6 Given the 2 x 2 algebraic system
Air = BC
8.11 Additional Symbols Introduced in Chapter 8 301
for which
where D(s) is det A. Prove that
det N = det A det B
that is, that (NI 1N22  N12N21) = D(B11B22  B12B21)
Relate this result to the elimination of f in (8.4,6).
8.7 Modify the system model of the altitudehold autopilot (Fig. 8.17) to include a rate term in the block indicated by the constant k. Show explicitly what changes result in (8.5,13).
8.8 Modify the analysis of Sec. 8.5 that leads to (8.5,13) to include climbing or gliding flight, that is, 9, f 0.
8.9 (a) Prove that in the yaw damper with washout, the steadystate yaw rate for a step in r,. is independent of the washout time constant and is given by
(b) Prove that if the washout filter is in the forward path, instead of the feedback path, then
regardless of the washout time constant.
8.10 A positive gain K in the p + 8, loop (Figs. 8.19 and 8.20e) implies right rudder re sponse when the right wing dips down. On the face of it this would seem to be desta bilizing. Explain, using the modal diagram Fig. 6.15 and Table 6.9, why the damping of the Dutch Roll mode is actually increased by this feedback. (Hint: r produced by a yaw damper is normally used to increase Dutch Roll damping.)
8.11 Add an outer loop to the system of Fig. 8.26 to control the heading angle I). Draw a new block diagram and write out the augmented system differential equation. (Hint: design the loop to command a bank angle proportional to heading error.)
8.12 Write out the full system of equations corresponding to (8.9,8).
8.13 (This is a miniresearch project). The relative locations of the pole P and the zero Z on Fig. 8.20f are relevant to the design of a yaw damper in that they determine the char acteristic structure of the root locus. Discover, by any means, what changes to the jet transport's aerodynamics would move its zero to the right of the pole. Two sugges tions: (1) Systematically perturb each of the stability derivatives and note the sensi tivity of the locations of P and Z to each, and (2) use approximate transfer functions to get analytical approximations for P and Z. When you have found some changes to the stability derivatives that would produce the desired result, discuss the design changes that would be needed to achieve the altered derivatives.
302 Chapter 8. Closed Loop Control
8.11 Additional Symbols Introduced in Chapter 8
control vector
denominator of transfer function
a ~ l a v
error vector
matrix of gust derivatives
gust vector
transfer function g =$ x
closedloop transfer function
feedback matrix
transfer function e 3 c gain constant
numerator of transfer function
gust gradient aw,/ay
gust gradient au,lay
gust gradient au,/ax
reference vector
gust input matrix
a ~ ~ a v speed at which T, = Dv
components of gust velocity
output vector
feedback vector
spectrum function
time constant
A P P E N D I X A
Analytical Tools
A.1 Linear Algebra
In this book no formal distinction is made between vectors and matrices, the former being simply column matrices, as is common in treatments of linear algebra. In par ticular the familiar vectors of mechanics, such as force and velocity, are simply three component column matrices. We use boldface letters for both matrices and vectors, for example, A = [a,,] and v = [v,]. The corresponding lowercase letter defines the magnitude (or norm) of the vector. The transpose and inverse are denoted as usual by superscripts, for example, AT and A'. When appropriate to the context, a subscript is used to denote the frame of reference for a physical vector, for example, V, = [u v wIT denotes a vector whose components in frame F, are (u, v , w). The three component vectors of physics have the following properties:
Scalar product
c is a scalar, with magnitude ab cos 6, where 6 is the angle between a and b
Vector product
where (A. 1,3)
c is a vector perpendicular to the plane of a and b, with direction following the righthand rule for the sequence a, b, c and has the magnitude ab sin 6, where 6 is the angle (< 180") between a and b
Unit vectors
The basis unit vectors are i,j,k such that
304 Appendix A. Analytical Tools
Where aij is the Kronecker delta.
Square matrices have the following properties:
Minor Determinant, Cofactor
The minor determinant mu of a matrix [a,,] is the determinant of the reduced ma trix that remains after the ith row and jth column of [a,,] have been deleted.
The cofactor is cij = mi,( l)"',
Adjoint
The adjoint of a matrix is the transpose of the matrix of cofactors,
adj [au] = [cuIT (A. 1,5)
Inverse
Provided that det A # 0 the inverse is given by
adj A A  1 = 
det A (A. 1,6)
A.2 The Laplace Transform
Let x(t) be a known function of t for values of t > 0. Then the Laplace transform of x(t) is defined by the integral relation
The integral is convergent only for certain functions x(t) and for certain values of s. The Laplace transform is defined only when the integral converges. This restriction is weak and excludes few cases of interest to engineers. It should be noted that the orig inal function x(t) is converted into a new function of the transform variable s by the transformation. The two notations for the transform shown on the lefthand side of (A.2,l) will be used interchangeably. The transforms of some functions that com monly occur in problems of linear systems are listed in Table A. 1.
TRANSFORMS O F DERIVATIVES
When xeCSt+ 0 as t 03 (only this case is considered), then
A.2 The Laplace Transform 305
Table A.1 Laplace Transforms
306 Appendix A. Analytical Tools
where x(0) is the value of x(t) when t = 0.' The process may be repeated to find the higher derivatives by replacing x(t) in (A.2,2) by k(t), and so on. The result is
TRANSFORM OF AN INTEGRAL
Let the integral be
and let it be required to find J(s). By differentiating with respect to t, we get
thus
and
METHODS FOR THE INVERSE TRANSFORMATION
The Use of Tables of Transforms
Extensive tables of transforms (like Table A.l) have been published that are use ful in carrying out the inverse process. When the transform involved can be found in the tables, the function x(t) is obtained directly.
The Method of Partial Fractions In some cases it is convenient to expand the transform Z(s) in partial fractions, so
that the elements are all simple ones like those in Table A. 1. The function x(t) can then be obtained simply from the table. This procedure is illustrated with an example. Let the secondorder system of Sec. 7.3 be initially quiescent, that is, x(0) = 0, and k(0) = 0, and let it be acted upon by a constant unit force applied at time t = 0. Then f (t) = 1, and f(s) = 11s (see Table A. 1). Then (see (7.4,l))
'To avoid ambiguity when dealing with step functions, t = 0 should always be interpreted as t = O+.
A.2 The Laplace Transform 307
Let us assume that the system is aperiodic; that is, that 5 > 1. Then the roots of the characteristic equation are real and equal to
A,,, = n 2 w' (A.2,6)
where
The denominator of (A.2,5) can be written in factored form so that
Now let (A.2,7) be expanded in partial fractions,
By the usual method of equating (A.2,7) and (A.2,8), we find
Therefore
By comparing these three terms with items 3 and 8 in Table A.l, we may write down the solution immediately as
Heaviside Expansion Theorem When the transform is a ratio of two polynomials in s, the method of partial frac
tions can be generalized. Let
308 Appendix A. Analytical Tools
where N(s) and D(s) are polynomials and the degree of D(s) is higher than that of N(s). Let the roots of D(s) = 0 be a,, so that
D(s) = (s  a,)@  a,) (s  a,)
Then the inverse of the transform is
The effect of the factor (s  a,) in the numerator is to cancel out the same factor of the denominator. The substitution s = a, is then made in the reduced expression.,
In applying this theorem to (A.2,7), we have the three roots a , = 0, a, = A , , a, = A,, and N(s) = 1. With these roots, (A.2,9) follows immediately from (A.2,lO).
The Inversion Theorem
The function x(t) can be found formally from its transform X(s) by the application of the inversion theorem Jaeger (1949) and Carslaw and Jaeger (1947). It is given by the line integral
x(t) =  lim es'X(s) ds 2 9 ~ i wffi C im
where y is a real number greater than the real part of all values of s for which X(s) di verges. That is, s = y is a straight line on the s plane lying parallel to the imaginary axis, and to the right of all the poles of X(s). This theorem can be used, employing the methods of contour integrals in the complex plane, to evaluate the inverse of the transform.
Extreme Value Theorems
Equation (A.2,2) may be rewritten as
T
= lirn Q e'%(t) dt Tm
We now take the limit s + 0 while T is held constant, that is,
T
x(O) + lirn s ~ ( s ) = lirn lirn ePs'i(t) dt s0 T+m s0
T
= lirn i(t) dt = lirn [x(T)  x(O)] Tm Tm
Hence lirn sX(s) = lim x(T) s0 Tm
This result, known as thefinal value theorem, provides a ready means for determin ing the asymptotic value of x(t) for large times from the value of its Laplace trans form.
'For the case of repeated roots, see Jaeger (1949)
A.3 The Convolution Integral 309
In a similar way, by taking the limit s + a at constant T, the integral vanishes for all finite x(t) and we get the initial value theorem.
lim sZ(s) = x(0) s+=
A.3 The Convolution Integral
The response of any linear system to any arbitrary input f ( t ) can be obtained from in tegrals of the two basic response functions h(t) and A(t). h(t) is the response to the unit impulse S( t ) , and A(t) is the response to the unit step l ( t ) . The system is assumed to be initially quiescent. If not, the transient associated with nonzero initial condi tions must be added to the following integrals. The response to f(t) is then given by Duhamel's integral, or the convolution integral:
When f(0) is not zero, then there must be added to (A.3,lb) a term to allow for the initial step in f ( t); i.e.,
x(t) = f(O)A(r) + A(r  ~ ) f ( r ) d~ 7=0
The physical significance of these integrals is brought out by considering them as the limits of the following sums
Typical terms of the summations are illustrated in Figs. A. 1 and A.2. The summation forms are quite convenient for computation, especially when the interval Ar is kept constant.
Figure A.1 Duhamel's integral, impulsive form. Ax = h(t  T ) ~ ( T ) AT = response at time t to impulse at time T.
310 Appendix A. Analytical Tools
 I . A " "Staircase" representation of f ( t l
Figure A.2 Duhamel's integral, indicia1 form. Af = step input applied at time 7, Ax = A(t  r)Af = response at time t to step input A f .
A.4 Coordinate Transformations
TRANSFORMATION OF A VECTOR
Let v be a vector with the components
The component of val in the direction of xbi is ual cos (Oil) where Oil denotes the an gle between O,xbi and Ouxal (see Fig. A.3). Thus by adding the three components of uaj in the direction of xbi we get
where
1, = cos (0,) (A.42)
are the nine direction cosines. (A.4,l) is evidently the matrix product
where
and constitutes the required transformation formula. Its inverse readily reverses the transformation to give
v , = LZV, = Labvb
A.4 Coordinate Transformations 311
where
Component of v,, on gi
X b i
4 3
Figure A.3 Component of vector.
(A.4,4) L,, = Lid
When a vector is successively transformed through several frames of reference, for example, Fa, F,, F,. . . . then
and
Since also v, = Lc,v,,, then it follows that
and similarly for additional transformations. The sequence of subscripts in the preceding expression should be noted, as it
provides a convenient mnemonic for remembering these relations.
PROPERTIES OF THE L MATRIX
Since v, and v, are physically the same vector v, the magnitude of v, must be the same as that of v,, that is, v 2 is an invariant of the transformation. From (A.4,3) this requires
It follows from the last equality of (A.4,5) that
Li,Lb, = I (A.46)
Equation (A.4,6) is known as the orthogonality condition on L , . From (A.4,6) it fol lows that
lL,,12 = 1
and hence that L , , is never zero and the inverse of L,, always exists. In view of (A.4,6) we have, of course, that
Li, = L,,' = L,, (A.4,7)
312 Appendix A. Analytical Tools
that is the inverse and the transpose are the same. Equation (A.4,6) together with (A.4,3b) yields a set of conditions on the direction cosines,
It follows from (A.4,8) that the columns of L,, are vectors that form an orthogonal set (hence the name "orthogonal matrix") and that they are of unit length.
Since (A.4,8) is a set of six relations among the nine lo, then only three of them are independent. These three are an alternative to the three independent Euler angles for specifying the orientation of one frame relative to another.
THE L MATRIX IN TERMS O F ROTATION ANGLES
The transformations associated with single rotations about the three coordinate axes are now given. In each case Fa represents the initial frame, Fb the frame after rota tion, and the notation for L identifies the axis and the angle of the rotation (see Fig. A.4). Thus in each case
By inspection of the angles in Fig. A.4, the following matrices are readily verified.
L,(xl) = [ C O X sinox]
0 sin X, cos X,
cos X, 0 sin X2
sinX, 0 cos X,
(4 Figure A.4 The three basic rotations. (a) About x,,. (b) About x,,. (c) About x,,.
A.4 Coordinate Transformations 313
The transformation matrix for any sequence of rotations can be constructed readily from the above basic formulas. For the case of Euler angles, which rotate frame FE into F, as defined in Sec. 4.4, the matrix corresponds to the sequence (X,, X,, X,) =
($, 8, 4), giving
LAE = LI(4) . . L3($) (A.4,I 1)
[The sequence of angles in (A.4,11) is opposite that of the rotations, since each trans formation matrix premultiplies the vector arrived at in the previous step.] The result of multiplying the three matrices is
TRANSFORMATION OF THE DERIVATIVE OF A VECTOR
Consider a vector v that is being observed simultaneously from two frames Fa and F, that have relative rotationsay F, rotates with angular velocity w relative to Fa, which we may regard as fixed. From (A.4,3)
(A.4,12) L,, =
The derivatives of v, and v, are of course
7
cos 8 cos $ cos 0 sin sin 0
sin 4 sin 8 cos $ sin 4 sin 8 sin $ sin q5 cos 8  cos 4 sin $ + cos 4 cos $
cos 4 sin 6 cos $ cos 4 sin 8 sin $ cos 4 cos 8 + sin 4 sin $ sin 4 cos $
where v,, = (d/dt)(u,,), and so forth. It is important to note that v, and v, are not simply two sets of components of the same vector, but are actually two difSerent vec tors.
Now because F, rotates relative to F,, the direction cosines lij are changing with time, and the derivative of (A.4,3) is
or alternatively
the second terms representing the effect of the rotation. Since L must be independent of v, the matrix L,, can readily be identified by
considering the case when v, is constant (see Fig. AS.). For then, from the funda mental definitions of derivative and cross product, the derivative of v as seen from F, is readily shown to be
314 Appendix A. Analytical Tools
Figure A.5 Rotating vector of constant magnitude.
The matrix equivalent to (A.4,15) is
where
0 a waz
The corresponding result from (A.4,14) is
va = Labvb (A.4,17)
It follows from equating (A.4,16) and (A.4,17) that
or
i abvb = haLabvb (A.4,18)
for all v,. Whence
iab = OaLab
and ha = L ~ ~ L ~ ~
Finally if the above argument is repeated with Fb considered fixed, and Fa having an gular velocity  u, we clearly arrive at the reciprocal result
Lba =  hbLba (A.4,19)
From (A.4,18) and (A.4,19), recalling that h is skewsymmetric so that OT =  0,  the reader can readily derive the result
A.5 Computation of Eigenvalues and Eigenvectors 315
From (A.4,14), (A.4,18), and (A.4,19) we have the alternative relations
with two additional permutations made possible by (A.4,20). A particular form we shall finally want for application is that which uses the components of v, transformed into F,, viz.
Lbc,va = v, + Obvt, (A.4,22)
TRANSFORMATION OF A MATRIX
Equation (A.4,20) is an example of the transformation of a matrix, the elements of which are dependent on the frame of reference. Generally the matrix of interest A oc curs in an equation of the form
v = Au (A.4,23)
where the elements of the (physical) vectors u and v and of the matrix A are all de pendent on the reference frame. We write (A.4,23) for each of the two frames F, and F,, that is,
and transform the second to
Lhavu = AbLbaua
Premultiplying by Lo, we get
Vii = LabAhLbuurr
By comparison with (A.4,24a) we get the general result
A, = La,A,Lba (A.4,26)
A.5 Computation of Eigenvalues and Eigenvectors
Some software packages provide for the calculation of eigenvalues and eigenvectors of matrices directly. The software used for many of the computations in this book is the Student Version of Program cC ,~ which does not do this. However, these impor tant system properties can readily be obtained from it, as shown in the following.
Program CC is oriented to the calculation of transfer functions and presents them in various forms; one is the polezero form. Any transfer function of the system, for example, that from elevator angle to pitch rate, when displayed in this form, will show the eigenvalues in the denominator. That is how we obtained the eigenvalues presented in Chap. 6.
'Available from Systems Technology, Inc., 13766 South Hawthorne Blvd., Hawthorne, CA, 90250 7083 U.S.
316 Appendix A. Analytical Tools
For the eigenvectors, we turn to the expansion theorem (A.2,10). Consider a case where the input to the system is 6, = 6(t), Dirac's delta function. The response of the ith component of the state vector to this input in the mode corresponding to eigen value A is
The ratio of this component to x , for the same input 6(t) is
This ratio gives the ith component of the eigenvector for the mode associated with A. Any component can be chosen for reference instead of x, , as illustrated in Figs. 6.3 and 6.15.
A.6 Velocity and Acceleration in an Arbitrarily Moving Frame
Since in many applications, we want to express the position, inertial velocity, and in ertial acceleration of a particle in components parallel to the axes of moving frames, we need general theorems that allow for arbitrary motion of the origin, and arbitrary angular velocity of the frame. These theorems are presented below.
Let F,(Oxyz) be any moving frame with origin at 0 and with angular velocity o relative to F,. Let r = r, + r' be the position vector of a point P of FM (see Fig. A.6). Let the velocity and acceleration of P relative to FI be v and a. Then in F,
We want expressions for the velocity and acceleration of P in terms of the compo nents of r' in FM. Expanding the first of (A.6,l)
v, = r,, + r; = v,, + r;
Figure A.6 Moving coordinate system.
A.6 Velocity and Acceleration in an Arbitrarily Moving Frame 317
where v, is the velocity of 0 relative to F,. The velocity components in FM are given
by V M = LMIv1 = LM/(vOI + rj) = VOM f LMIrj
From the rule for transforming derivatives (A.4,22)
whence
vM = v,, + rL + OM&
The first term of (A.6,4) is the velocity of 0 relative to F,, the second is the velocity of P as measured by an observer fixed in FM, and the last is the "transport velocity," that is the velocity relative to F, of the point of FM that is momentarily coincident with P. The total velocity of P relative to FI is the sum of these three components. Following traditional practice in flight dynamics, we denote
(When necessary, subscripts are added to the components to identify particular mov ing frames.)
The scalar expansion of (A.6,4) is then
u, = u,, + x + qz  ry
u, = u,,, + j + rx  p z
These expressions then give the components, parallel to the moving coordinate axes, of the velocity of P relative to the inertial frame.
On differentiating v, and using (A.6,4) we find the components of inertial accel eration parallel to the FM axes to be
where a,, = v,, + hMv,,, = LM&, is the acceleration of 0 relative to F,. The total inertial acceleration of P is seen to be composed of the following parts:
a,,: the acceleration of the origin of the moving frame
rL: the acceleration of P as measured by an observer fixed in the mov ing frame
r : the "tangential" acceleration owing to rotational acceleration of the frame FM
2hMr,&: the Coriolis acceleration
O,hMrL: the centripetal acceleration
Three of the five terms vanish when the frame FM has no rotation, and only rL re mains if it is inertial. Note that the Coriolis acceleration is perpendicular to wM and
318 Appendix A. Analytical Tools
rL, and the centripetal acceleration is directed along the perpendicular from P to w. The scalar expansion of (A.6,7) gives the required inertial acceleration components of P as
A P P E N D I X B
Data for Estimating Aerodynamic Derivatives
This appendix contains a limited amount of data on stability and control derivatives. It is not intended to be used for design. That requires much more detail than could possibly be provided here. It is intended to display some representative orders of magnitude and trends, and to provide numerical data that teachers and students can use for exercises. All the data pertain to subsonic flight of rigid airplanes. Much of the information comes from either the USAF Datcom (USAF, 1978) or from the data sheets of the Royal Aeronautical Society of Great Britain (now out of print), which is also the source for some of the Datcom data. We have taken some liberties in extract ing and presenting this information, but have not changed any essential content. For information about derivatives at transonic and supersonic speeds and for geometries different from those covered in the following, the reader is referred to the USAF Dat com. When estimating derivatives, reference should also be made to Tables 5.1 and 5.2.
B.1 LiftCurve Slope, CLa
B.2 Control Effectiveness, C,,
B.3 Control Hinge Moments
B.4 Tab Effectiveness, b,
d€ B.5 Downwash,  acu
B.6 Effect of Bodies on Neutral Point and Cmo
B. 7 Propeller and Slipstream Effects
B.8 Wing Pitching Derivative, Cm4
B.9 Wing Sideslip Derivatives C,,, C,,,
B. 10 Wing Rolling Derivatives ClP, C,
B.11 Wing Yawing Derivatives C,,, Cnr
B.12 Changes in Inertias and Stability Derivatives with Change of Body Axes
320 Appendix B. Data for Estimating Aerodynamic Derivatives
B.1 LiftCurve Slope, C,. NOTATION
aspect ratio
chord
liftcurve slope of wing alone
twodimensional (airfoil) liftcurve slope
theoretical value of Cla
an empirical factor
Mach number
Reynolds number, Vcplp
PrandtlGlauert compressibility factor,
P C l P sweepback angle of midchord line
NOTES
The source of the data for airfoils and wings is USAF datcom. It applies to rigid straighttapered wings at subsonic speeds and small angle of attack.
The section liftcurve slope is given by
where K is given in Fig. B.l , la and (C,,),,,,, in Fig. B.l,lb. Y,, and Y,, are the air foil thicknesses, in percent of chord, at 90% and 99% of the chord back from the leading edge, as illustrated, and the trailing edge angle is defined in terms of these thicknesses by
%(y90  y99) tan %& =
The liftcurve slope Cia of the wing alone is given in Fig. B.1,2. The inset equation is seen to approach the theoretically correct limits of vA/2 as A  0 and2.rras { A  m , K l , A  O , p + 1) .
Figure B. 1,3 gives some theoretical values of the body effect on CLa for unswept wings in midwing combination with an infinite circular cylinder body. For values of A < 1, the theory also applies to delta wings with pointed tips.
In Fig. B.l,3a the wing angle of attack is the same as that of the fuselage; that is, E = 0. In this case the lift of the wingbody combination increases to a maximum value, then decreases with increasing body diameter. Where there is a wing setting, i.e., E # 0, and cu, = 0 (Fig. B.1,3b), the lift of the combination decreases with in creasing a.
B.2 Control Effectiveness, C,, 321
 Mean lhne
Wing th~ckness ratlo, tlc
(b )
Figure B.1,l Twodimensional liftcurve slope.
B.2 Control Effectiveness, C,,
SECTION DATA
Figure B.2,la presents theoretical values of the twodimensional control derivative C,, for simple flaps in incompressible flow. These values can be corrected by the em
322 Appendix B. Data for Estimating Aerodynamic Derivatives
 A , [p2+ tan2 A , , ~ ] ~
Figure B.Z,2 Subsonic wing liftcurve slope.
pirical data of Fig. B.2,lb for the strong effect of nonideal liftcurve slope of the main surface to which the control is attached.
SURFACE DATA
The derivative C,, for a finite lifting surface with a part span control flap is obtained from the section derivative by
c,, = c,, (4) K l K* Cia
where CLa and CIa are as defined in B.l, C,, is the corrected value from Fig. B.2,lb and K, and K, are the factors given in Figs. B.2,2 and B.2,3. In these figures the para meter (a,),, is the rate of change of zerolift angle with flap deflection, given by the inset graph, and A, and A are, respectively, the aspect ratio and taper ratio of the main surface.
B.3 Control Hinge Moments
NOTATION
7 trailingedge angle defined by the tangents co the upper and lower surfaces at the trailing edge
theoretical rate of change of hingemoment coefficient with angle of attack for incompressible inviscid twodimensional flow
actual rate of change of hingemoment coefficient with angle of attack for incompressible twodimensional flow
B.3 Control Hinge Moments 323
Body diameter to wing span, a = d/b
(a )
A€ Rectangular wings infinite circular cylinder midwing configuration
Body diameter to wing span, o = d/b ( b )
Figure B.1,3 Body effect on liftcurve slope expressed as a ratio of lift of wingbody combination to lift of wing alone. (From "Lift and Lift Distribution of Wings in Combination with Slender Bodies of Revolution," by H. J. Luckert, Can. Aero. J., December 1955.)
324 Appendix B. Data for Estimating Aerodynamic Derivatives
(C1b)theory
(per radian)
Figure B.51 Control effectiveness for twodimensional incompressible flow. (From Royal Aeronautical Society Data Sheet Controls 0 1.0 1.03.)
(b2)0T theoretical rate of change of hingemoment coefficient with control deflection for incompressible inviscid twodimen sional flow
@2)0 actual rate of change of hingemoment coefficient with con trol deflection for incompressible twodimensional flow
B.3 Control Hinge Moments 325
1 0 0 2 4 6 8 10
A,
Figure B.52 Flapchord factor.
F , , a i /6
( b ~ )obaI9 (b,)obal rates of change of control hingemoment coefficients with in cidence and controlsurface deflection, respectively, in two dimensional flow for control surfaces with sealed gap and nose balance
induced angle of attack correction to (b,), and (b,),, respec tively, where F, is the value of (ai/6) [CdCla] when cf = c
streamline curvature correction to (b,), and (b,),, respec tively, where F, is the value of A(b,) when cf = c
F3
Balance
factor to F, and A(b,) allowing for nose balance
ratio of controlsurface area forward of hinge line to control surface area behind hinge line
NOTES
Figures B.3,l and B.3,2
The curves of Fig. B.3,1 were derived for a standard series of airfoils with plain controls for which tan (4)~ = t/c (referred to by an asterisk). To correct for airfoils
326 Appendix B. Data for Estimating Aerodynamic Derivatives
0 0 .2 .4 .6 .8 1 .O
= Y I ~ 2
Figure B.2,3 Span factor for inboard flaps.
with tan (+)T different from t/c, values of (b , ) ; , (Cla):he,, and Cya are calculated for the given t/c ratio; then (b,), is calculated from
Values of (Cl,)~heo, and CTm may be obtained as in Appendix B. 1. The curves apply for values of angle of attack and control deflection for which
there is no flow separation over the airfoil; for these conditions (b,), can be estimated to within 20.05. The data refer to sealed gaps but may be used if the gap is not greater than 0.002~.
The above discussion also applies to the data given in Fig. B.3,2 for (b,),. The subscript 1 in Eq. B.3,l becomes a subscript 2, a becomes 6, and values of (C,,),",,,, and C;, may be obtained from Appendix B.2.
B.3 Control Hinge Moments 327
0 0.1 0.2 0.3 0.4
"fk
Figure B.3,l Rate of change of hingemoment coefficient with angle of attack for a plain control in incompressible twodimensional flow. (From Royal Aeronautical Society Data Sheet Controls, 04.01.01.)
Figure B.3,3
The effect of nose balance on (b,), and (b,), can be estimated from the curves given on this figure. The data were obtained from windtunnel tests on airfoils with controlchordlairfoilchord ratio of 0.3. Relatively small changes in nose and trailing edge shape, and airflow over the control surface, may have a large effect on hinge moments for balanced control surfaces, so that estimates of nosebalance effect will be fairly inaccurate. If the controlsurface gap is unsealed, the hingemoment coeffi cients of plain and nosebalanced controls will generally become more positive.
328 Appendix B. Data for Estimating Aerodynamic Derivatives
0 0.1 0.2 0.3 0.4
cr/"
Figure B.3,2 Rate of change of hingemoment coefficient with control deflection for a plain in twodimensional flow. (From Royal Aeronautical Society Data Sheet Controls 04.01.02.)
Figure B.3,4 Twodimensional hingemoment coefficients for control surfaces with nose bal
ance can be corrected for finite aspect ratio of the main surface using the factors given in the curves and the following equations:
B.3 Control Hinge Moments 329
0 0.1 0.2 0.3 0.4 %0.5 0.6 0.7 Balance ratio = [(cb/cf)'  (t/2cf)' ] ' I 2
" 0 1.0 0.2 0.3 0.4 0.5 0.6 0.7
Balance ratio
Figure B.3,3 Effect of nose balance on twodimensional plaincontrol hingemoment coefficients. (From Royal Aeronautical Society Data Sheet 04.01.03.)
For plain control surfaces the above equations are used with F, = 1. (b,), and (b,), can be obtained from Fig. B.3,1 and B.3,2, respectively, for plain controls. For nose balanced controls, the twodimensional coefficients (b,), and (b,), must include the effect of nosebalance. Values of C,m can be obtained from Sec. B. 1, and those for C,, from Fig. B.2,l.
Liftingsurface theory was applied to unswept wings with elliptic spanwise lift distribution to derive the factors. Fullspan control surfaces were assumed together with constant ratios of cflc and constant values of (b,), and (b,), across the span. The factors apply to wings with taper ratios of 2 to 3 if cflc, (b,), and (b,), do not vary by more than + 10% from their average values.
330 Appendix B. Data for Estimating Aerodynamic Derivatives
Figure B.3,4 Finiteaspectratio corrections for twodimensional plain and nosebalanced control hingemoment coefficients (C, per radian). (From Royal Aeronautical Society Data Sheet Controls 04.01.05.)
B.4 Tab Effectiveness, b,
The data and method that follows is taken from the USAF Datcom. It provides esti mates of b, [see (2.5,1)] for twodimensional subsonic attached flow over airfoils with a control surface and tab. Corrections for partspan tabs can be made by multi plying the result for two dimensions by the ratio of the control surface area spanned by the tab to the total control surface area (both areas being measured aft of the hinge line).
B.4 Tab Effectiveness, b, 331
where
( ) is the change in control section hingemoment coefficient due to tab cl.61 deflection, measured at constant values of lift and flap deflection.
This value is obtained from Fig. B.4,l.
is the change in control section hingemoment coefficient due to lift i2)..ll variation, measured at constant values of tab and flap deflection.
This value is obtained from Fig. B.4,2.
is the section liftcurve slope of the primary panel (wing, horizontal (')al,~ tail, etc.) at constant values of tab and flap deflection. This value
can be obtained from (B. 1,l).
is the rate of change of angle of attack due to a change in tab deflec i g ) c 1 3 a j tion in the linear range at constant values of lift and flap deflection.
This value can be obtained from Fig. B.4,3.
.004
F& 
Expenmental (NACA 0009 a~rfo~lround nose, sealed gaps)
I I I I I I I 0 2 .4 .6 8 1 0
cf/c
Figure B.4,1 Effect of tab deflection on controlsurface section hinge moments.
332 Appendix B. Data for Estimating Aerodynamic Derivatives
0 .2 .4 .6 .8 1 .o cf/c
Figure B.4,2 Effect of section lift coefficient on flap section hinge moments.
d€ B.5 Downwash,  aa
The method and data that follow are taken from the USAF Datcom. The average low speed downwash gradient at the horizontal tail is given by
Figure B.4,3 Rate of change of angle of attack due to a change in tab deflection.
a€ B.5 Downwash,  333 aa
.5
.4
.3
KA
.2
1
I I I I I I I I I
O o l l l l l l l l l 2 4 A 6 8 10
Figure B.S,l Wing aspectratio factor.
where KA, KA, and K, are wingaspectratio, wingtaperratio, and horizontaltaillo cation factors obtained from Figs. B.5,1, B.5,2, and B.5,3, respectively. A,, is the sweepback angle of the wing $ chord line.
At higher subsonic speeds the effect of compressibility is approximated by
334 Appendix B. Data for Estimating Aerodynamic Derivatives
Tall mac
1 ,
 Wing \ ~ o o t chord f
Figure B..5,3 Horizontaltaillocation factor.
where
(' bw speed
is obtained using (B.5,l)
(CL,)low speed and (C,,), are the wing liftcurve slopes at the appropriate Mach numbers, obtained by using the straighttaperedwing method of Sec. B. 1
B.6 Effectof Bodies on Neutral Point and C,, 335
B.6 Effect of Bodies on Neutral Point and C,,
NOTATION
local wing chord at center line of fuselage or nacelle
mean aerodynamic chord
maximum width of fuselage or nacelle
gross wing area
shift of neutral point due to fuselage or nacelle as a fraction of 2, positive aft
area of planform of body
area of planform of body, forward of 0.252
root chord of wing without fillets
increment to C,, due to a body at zero lift
reflex angle of fillet, i.e., angle between wing root chord and lower surface of fillet for upswept fillets, or the upper surface for down swept fillets, positive as indicated in Fig. B.8,2
fillet liftincrement ratio, i.e., C,,/C,m, considering the fillet to be a flap of chord 1
NOTES
Figure B.6,l: Ah,
The data for estimating Ah, presented in this graph were derived from windtun nel tests. The forward shift in neutral point is mainly dependent on the length and width of the body forward of the wing. The values of Ah, given by the curves are ac curate to within %0.01c, and are about 5% higher for lowwing, and the same amount lower for highwing configurations. The data are inapplicable if the wing is clear of the body. Separate values should be computed for fuselages and nacelles, and the re sults added to obtain the total neutralpoint shift.
Figure B.6,2: (C,,),
The curves given in this figure apply to streamline bodies of circular or near cir cular cross section with midwing configurations. For high or lowwing configura tions a positive or negative A(Cm,,), = 0.004 is added, respectively, to the value de rived from the curves. The curves apply only for angles of attack up to about 15" for streamline bodies where the pitching moment of the body varies linearly with angle of attack.
In the windtunnel tests from which the data were derived, the wings had straight trailing edges at the wingbody junction. Fillets have a large effect on C,,,, however, especially if 0 is large. The following equation may be used to estimate the fillet ef fect if 0.12 < l,,lc < 0.5 and 0.03 < Sflb < 0.075:
336 Appendix B. Data for Estimating Aerodynamic Derivatives
c/l
Figure B.6,Z Effect of a fuselage or nacelle on neutralpoint position. (From Royal Aeronautical Society Data Sheet Aircraft 08.01 .01.)
C,, due to fillets = [0.046 + 0.08(dC,JdCL),Ae  0.2(c + lf)/c](w + Sf)lb
The value obtained from the curves, the fillet effect, and the effect due to wing posi tion are added to determine (C,,),.
B. 7 Propeller and Slipstream Effects
PROPELLER NORMAL FORCE
The following method of estimating the propeller normal force is due to Ribner (1944). The normal force is expressed in terms of the derivative a C N p a P (see Sec. 3.4), which is given by
The factor f is the same for all propellers, and is given in Fig. B.7,l as a function of T, = T / ~ v ~ ~ ~ . The value of CYho varies with the propeller and its operating condition. The values for a particular propeller family are given in Fig. B.7,2. Extrapolation to
8.7 Propeller and Slipstream Effects 337
,Wing nolift line
Zero pitchingmoment line of body alone 
Mean aerodynamic quarterchord position
I
Fillet
0.30
0.40
0.25 0.35
0.30
0.20 0.25
I,, 55 42 0.20 0.15
uE .... 0.15  'I 0.10 0.10
0.05 0.05
0 0 0.2 0.4 0.6 0.8 1 .O
SBF 'BF SB 'B
Figure B.6,2 Effect of a fuselage on C,,. (From Royal Aeronautical Society Data 08.01.07.)
Sheet Aircraft
other propellers can be made by means of Fig. B.7,3, on the basis of the "sideforce factor," SFF. This is a geometrical propeller parameter, given approximately by
SFF = 525[(blD),., + (blD)o,6] + 270(b/D),.,
where (b/D) is the ratio of blade width to propeller diameter, and the subscript is the relative radius at which this ratio is measured.
Also given in Ribner (1944) are some curves which are useful for estimating the upwash or downwash at the propeller plane. These are reproduced in Fig. B.7,4.
338 Appendix B. Data for Estimating Aerodynamic Derivatives
=c
Figure B.7,I Variation of f with T,. (From NACA Wartime Rept. L25, 1944, by H. S. Ribner.)
LIFT DUE TO SLIPSTREAM
The method of Smelt and Davies (1937)* can be used to estimate the added wing lift due to the slipstream. It is given by
where
Dl = diameter of slipstream at the wing C.P.
= D[(1 + a)l(l + s)]lt2 c = wing chord on center line of slipstream
Figure B. 7,2 S . Ribner.)
"ee also Ribner and Ellis (1972).
B.7 Propeller and Slipstream Effects 339
Distance behind root quarterchord point, root chords
Figure B.7,4 Value of 1  d d d a on longitudinal axis of elliptic wing for aspect ratios 6,9, and 12. (From NACA Wartime Rept. L25, 1944, by H. S. Ribner.)
340 Appendix B. Data for Estimating Aerodynamic Derivatives
0 2 4 6
Aspect ratio of wing portion in slipstream
Figure 8.73 Empirical factor A. (From Smelt and Davies, 1937.)
S = wing area
s = a + a x l ( ~ ~ l 4 + A?)'/~
D = propeller diameter a = 1. 2 + K1 + ~ T , / T ) " ~
x = distance of wing C.P. behind propeller
C,, = lift coefficient at section on slipstream center line, in absence of the slip stream
a, = twodimensional liftcurve slope of wing section
8 = angle of downwash of slipstream at wing C.P. calculated from the equa tion
1/8°.8 = 0.016xlD + l/800.8
where
8, = a+l(l + a )
4 = angle between propellor axis and direction of motion.
A is an empirical constant given in Fig. B.7,5.
B.8 Wing Pitching Derivative Cm4
The method of USAF Datcom for estimating this derivative for a rigid wing in sub sonic flow is as follows. The lowspeed value (M = 0.2) of Cmu is given by
0.7C,, cos Acf4 A + 2 cos A,,
B.9 Wing Sideslip Derivatives CiB, CnP 341
where
C is the wing section lift curve slope from Sec. B. 1 (per rad).
A,,, is the sweepback angle of the wing chord line.
For higher subsonic speeds the derivative is obtained by applying an approximate  compressibility correction.
r a3 tan2 A,, +  1
I AB + 6 cos A,.,, (Cm,)~>0.2 = A"an2 A,, (C,n,,)M=0.2 (B.82)
+ 3 A + 6 cos A,,,
where A is aspect ratio, and
B.9 Wing Sideslip Derivatives C,,, C,,,
The methods that follow are simplified versions of those given in USAF Datcom. They apply to rigid straighttapered wings in subsonic flow.
The derivative C,,: For A 2 1 .O:
AC,, ..ID = cL[( %) K ~ , + (2)IA + I' (% K ~ ~ . ) + B tan A., 8 tan A,, (per deg)
CL A,,,
For A < 1 .O:
where
el*<l2 is the wingsweep contribution obtained from Fig. B.9,l.
K~~ is the compressibility correction to the sweep contribution, obtained from Fig. B.9,2.
is the aspectratio contribution, including taperratio effects, ob ( A tained from Fig. l3.9,3.
5 is the dihedral effect for uniform geometric dihedral, obtained from I' Fig. B.9,4.
r is the dihedral angle in degrees.
KM,. is the compressibility correction factor to the uniformgeometricdi hedral effect, obtained from Fig. B.9,5.
342 Appendin B. Data for Estimating Aerodynamic Derivatives
.010
Figure B.9,l Wing sweep contribution to C,,.
AC, is the wingtwist correction factor, obtained from Fig. B.9,6.
8 tan A,,
8 is the wingtwist between the root and tip stations, negative for washout (see Fig. B.9,6).
A12 is the sweepback angle of the midchord line.
A, is the sweepback angle of the f chord line.
344 Appendix B. Data for Estimating Aerodynamic Derivatives
.0002
c43  r
(per deg2)
.0001
0 2 4 6 8 10 Aspect ratio, A
C,p r
(per deg2)
.0002
c93  r
(per deg2)
0 2 4 6 8 10 Aspect ratio. A
0 2 4 6 8 10 Aspect ratio, A
Figure B.9,4 Effect of uniform geometric dihedral on wing Clp.
A2B2 + 4AB cos Ac14  8 C O S ~ Ac14 )( % ) A2 + 4A cos ACl4  8 cos2 AcI4 CL low speed
(B.994)
where B is given by (B.8,3).
B.10 Wing Rolling Derivatives C,p, CnP 345
M COS Ad2
Figure B.93 Compressibility correction to dihedral effect on wing C,p.
B. 10 Wing Rolling Derivatives Clp, CnP
The following methods are simplified versions of those given in USAF Datcom. They apply to rigid straighttapered wings in subsonic flow, in the linear range of C, vs. a.
The derivative Clp:
" 0 2 4 6 8 10 12 14
Aspect ratlo. A
Figure B.9,6 Effect of wing twist on wing Crp.
346 Appendix B. Data for Estimating Aerodynamic Derivatives
where
is the rolldamping parameter at zero lift, obtained from Fig. C ~ = O B. 10,l as a function of hp and PAIK.
The parameter K is the ratio of the twodimensional liftcurve slope at the appropriate Mach number to 2.rrlP; that is, (Cla),I(2?rIP). The twodimensional liftcurve slope is obtained from Sec. B. 1. For wings with airfoil sections varying in a reasonably linear man ner with span, the average value of the liftcurve slopes of the root and tip sections is adequate. The parameter Ap is the compressible sweep parameter given as
hp = tan' j T ) tan 'c" , where /3 = m. and A,, is the sweepback angle of the wing chord line.
(Cl,,), is the dihedraleffect parameter given by
(C I p ) r = o
where
r is the geometric dihedral angle, positive for the wing tip above the plane of the root chord.
Ap (deg)
Figure B.lO,la Roll damping C,", part 1.
B.10 Wing Rolling Derivatives C,p, CnP 347
( b ) h = 0.50
U
20 0 2 0 40 60 80 Ap (deg)
Figure B.lO,lb Roll damping Cll1, part 2.
Figure B.lO,lc Roll damping C,,,, part 3 .
348 Appendix B. Data for Estimating Aerodynamic Derivatives
z is the vertical distance between the CG and the wing root chord, positive for the CG above the root chord.
b is the wing span.
(AC,Jdra, is the increment in the rolldamping derivative due to drag, given by
where
(CISc~L is the dragduetolift rolldamping parameter obtained from Fig. C: B.10.2 as a function of A and A,.
CI. is the wing lift coefficient below the stall.
C,, is the profile or total zerolift drag coefficient.
The derivative C,:
Cnp =  CIp tan a  0 (B.10,4)
where
CIP is the rolldamping derivative at the appropriate Mach number esti mated above
a is the angle of attack.
CL. is the lift coefficient.
is the slope of the yawing moment due to rolling at zero lift given by M
t 1 A
Figure B.10,2 Dragduetolift rolldamping parameter.
B.10 Wing Rolling Derivatives C,p, CnP 349
1 AB +  (AB + cos A,,) tan2 AcI4
A + 4 cos A,,4 2
AB + 4 cos A,, 1 (2)cL=o ,%=o 1 A +  (A + cos A,.,) tan2 A,, 2
(B.10,5)
Aspect ratlo, A
Figure B.10,3 Effect of wing twist on wing rolling derivative Cnp.
350 Appendix B. Data for Estimating Aerodynamic Derivatives
where
B is given by (B.8,3).
Ad4 is the sweepback angle of the i chord line.
is the slope of the lowspeed yawing moment due to rolling at zero )' lift aven by ? r
tan A, + tan' A, A + 6(A + cos A,) ( h  h) 7
12
A + 4 cos A,
(B. 10,6)
ACnp is the effect of linear wing twist obtained from Fig. B.10,3.
0 0 is the wing twist between the root and tip stations in degrees, nega
tive for washout (see Fig. B. 10,3).
B.11 Wing Yawing Derivatives Cl,, C,,,
The following methods are simplified versions of those given in USAF Datcom. They apply to rigid straighttapered wings in subsonic flow at low values of C,.
The derivative ClP:
where
is the slope of the rolling moment due to yawing at zero lift given by M
A(l  B ~ ) AB + 2 cos A,, tan2 A,, 1 + +
 2B(AB + 2 cos A,) AB + 4 cos A,, 8 (2IcL M =o  A + 2 cos A, tan2 A,,
1 + A + 4 cos A,, 8 (2)cL=o M=O
(B.I1,2)
where
B is given by (B.8,3).
is the slope of the lowspeed rolling moment due to yawing at zero C, CL'O 1. ( ) M ift, obtained from Pig. B.1 l , l as a function of aspect ratio, sweep of
the quarterchord, and taper ratio. (B.11,2) modifies the lowspeed value by means of the PrandtlGlauert rule to yield approximate cor rections for the firstorder threedimensional effects of compressible flow up to the critical Mach number.
A Clr r is the increment in C,, due to dihedral, given by
A 1 T A sin A,,4    
r 12 A + 4 cos A,.,4
I. is the geometric dihedral angle in radians, positive for the wing tip above the plane of the root chord.
 is the increment in Clr due to wing twist obtained from Fig. B. 11,2. 8
8 is the wing twist between the root and tip sections in degrees, nega tive for washout (see Fig. B.11,2).
The derivative C,8r:
352 Appendix B. Data for Estimating Aerodynamic Derivatives
ACl,. 
(per deg)
where
C, is the wing lift coefficient.
 Cnr is the lowspeed dragduetolift yawdamping parameter obtained from C: Fig. B.11,3 as a function of wing aspect ratio, taper ratio, sweepback, and
CG position.
C, is the lowspeed profiledrag yawdamping parameter obtained from Fig. 8.1 1,4 as a function of the wing aspect ratio, sweepback, and CG posi tion.
C,, is the wing profile drag coefficient evaluated at the appropriate Mach number. For this application CD, is assumed to be the profile drag associ ated with the theoretical ideal drag due to lift and is given by
where CD is the total drag coefficient at a given lift coefficient.
B.12 Changes in Inertias and Stability Derivatives with Change of Body Axes 353
cn;cL2
Figure B.11,3 Lowspeed dragduetolift yawdamping parameter.
B.12 Changes in Inertias and Stability Derivatives with Change of Body Axes
A matrix A that connects two vectors u and v as in (A.4,23) transforms between two reference frames as in (A.4,26). We now apply this rule to the inertia matrix and to matrices of stability derivatives.
354 Appendix B. Data for Estimating Aerodynamic Derivatives
Aspect ratio, A
Aspect ratio. A
B.12 Changes in Inertias and Stability Derivatives with Change of Body Axes 355
TRANSFORMATION OF INERTIAS
The inertia matrix I connects angular momentum with angular velocity [see (4.3,4) and (4 .331 via
and hence belongs to the class of matrices covered by (A.4,26). It follows that for two sets of body axes, denoted F,, and F,, connected by the transformation LIZ, the inertias in frame F,, can be obtained from those in F,, by
If the two frames are two sets of body axes such that x,, is rotated about y,, through angle 5 to bring it to x,,, then (see Appendix A.4)
cos 5 0 sin 5 L 2 1 = [ o 1 0 1 (B.12,2)
sin 5 0 cos 5
The inertias in frame F,,, denoted by an asterisk, are then obtained from those in FB,, with the usual assumption of symmetry about the xz plane, by the relations
I]T = Ix cos2 5 + IZ sin2 5 + I, sin 25
I: = I, sin2 5 + IZ cos2 5  I= sin 25 (B.12,3)
I: = $(I,  I,) sin 25 + I, (sin2 6  cos2
TRANSFORMATION OF STABILITY DERIVATIVES
All of the stability derivatives with respect to linear and angular velocities and veloc ity derivatives can be expressed as sums of expressions of the form of (A.4,23). That is, with the usual assumptions about separation of longitudinal and lateral motion, we can write
0 0
(B. 12,5)
Each of the six matrices of derivatives above transforms according to the rule (A.4,26). When L is given by (B.12,2) we have the transformation from an initial set of body axes (unprimed) to a second set (primed) as follows:
356 Appendix B. Data for Estimating Aerodynamic Derivatives
Longitudinal
(Xu)' = Xu cos3 5  (X, + Z,) sin 5cos 6 + Z, sin2 5 (X,)' = X, cos2 ( + (Xu  Z,) sin 5 cos 5  Z, sin2 5 (X,)' = Xq cos 5  Zq sin 6 (X,)' = Z,sin2 5 (1)
(X,)' = Z, sin (cos 5 (1)
(Z,)' = Z, cos2 5  (Z,  Xu) sin 5 cos 5  X, sin2 5 (Z,)' = Z, cos2 5 + (Z, + X,) sin 5cos 5 + Xu sin2 6 (Z,)' = Zq cos 5 + Xq sin 5 (Z,)' = ZG sin 5 cos 6 (1) (Z*)' = z, c0s2 5 (Mu)' = Mu cos 6  M, sin 6 (M,)' = M, cos 6 + Mu sin (
(Mq)' = Mq (M,)' = M,sin( ( 1 )
(M,)' = MG cos ( ( 1 )
(B. 12,6)
Lateral
(Y,)' = y, (Y,)' = Yp cos 5  Y, sin 5 (Y,)' = Yr cos ( + Yp sin 5 (L,)' = L, cos 6  Nu sin 6 (L,)' = Lp cos2 5  (L, + N,) sin 5 cos 5 + N, sin2 (
(L,)' = L, cos2 5  (N,  Lp) sin (cos 6  N, sin2 6 (B. 12,7)
(N,)' = N, cos 5 + L, sin 6 (N,)' = Np cos2 5.  (N,  Lp) sin (cos 5  L, sin2 6 (N,)' = N, cos2 5 + (L, + N,,) sin 6 cos 5 + L, sin2 5
(1) For consistency of assumptions, the derivatives with respect to ti and (X,)' are usually ignored.
A P P E N D I X C
Mean Aerodynamic Chord, Mean Aerodynamic Centec
and C, acw
C.1 Basic Definitions
In the normal flight range, the resultant aerodynamic forces acting on any lifting sur face can be represented as a lift and drag acting at the mean aerodynamic center (1;. JJ,
2). together with a pitching couple C,,,e,c,L which is independent of angle of attack (see Fig. 2.8).
The pitching moment of a wing is nondimensionalized by the use of the mean aerodynamic chord C.
Both the m.a. center and the m.a. chord lie in the plane of symmetry of the wing. However, in determining them it is convenient to work with the halfwing.
These quantities are defined by (see Fig. C.1)'
2 "I2
C = s c 2 d y (C.l, l)
2  X ==  C,<p d y (C. 12) CLS
2 b 7 "  CLS o (C. 1,3)
(C. 1,4)
where b = wing span
c = local chord
C , = total lift coefficient
C , = local additional lift coefficient, proportional to C,~
C , = local basic lift coefficient, independent of C,
C, = C,, + C,<< = total local lift coefficient
'The coordinate system used applies only to this appendix.
358 Appendix C. Mean Aerodynamic Chord, Mean Aerodynamic Center and Cmncw
'\((\ centers
J. Figure C.1 Local aerodynamic center coordinates.
ma, = pitching moment, per unit span, about aerodynamic center (Fig. C.4)
S = wing area
y = spanwise coordinate of local aerodynamic center measured from axis of symmetry
x = chordwise coordinate of local aerodynamic center measured aft of wing apex
z = vertical coordinate of local aerodynamic center measured from xy plane
77,p = lateral position of the center of pressure of the additional load on the half wing as a fraction of the semispan
The coordinates of the m.a. center depend on the additional load distribution; hence the position of the true m.a. center will vary with wing angle of attack if the form of the additional loading varies with angle of attack. For a wing that has no aerodynamic twist, the m.a. center of the halfwing is also the center of pressure of the halfwing. If there is a basic loading (i.e., at zero overall lift, due to wing twist), then (X, 7, 2) is the center of pressure of the additional loading.
The height and spanwise position of the local aerodynamic centers may be as sumed known, and hence J and 2 for the halfwing can be calculated once the addi tional spanwise loading distribution is known. However, in order to calculate X, the foreandaft position of each local aerodynamic center must be known first. If all the local aerodynamic centers are assumed to lie on the nthchord line (assumed to be straight), then
where c, = wing root chord
A, = sweepback of nthchord line, degrees
Ideal twodimensional flow theory gives n = for subsonic speeds and n = 4 for su personic speeds.
C.2 Comparison of m.a. Chord and m.a. Center 359
The m.a. chord is located relative to the wing by the following procedure:
1. In (C. 1,2) replace C,<, by C,, and for x use the coordinates of the :chord line.
2. The value of 3 so obtained (the mean quarterchord point) is the fpoint of the m.a. chord.
The above procedure and the definition of c (see C. 1,l) are used for all wings.
C.2 Comparison of m.a. Chord and m.a. Center for Basic Plan forms and Loading Distributions
In Table C.l taken from (Yates, 1952), values of m.a. chord and 7 are given for some basic planforms and loading distributions.
In the general case the additional loading distribution and the spanwise centerof pressure position can be obtained by 'methods such as those of De Young and Harper (1948), Weissinger (1947), and StantonJones (1950). For a trapezoidal wing with the local aerodynamic centers on the nthchord line, the chordwise location of the mean aerodynamic center from the leading edge of the m.a. chord expressed as a fraction of the m.a. chord h,,, is given by
h,,,, = n +  1 A tan A, (C.2,1)
Table C.1
Additional Loading M.A. C.
Planform   Distribution c Y
Constant taper and sweep (trapezoidal)
Constant taper and sweep (trapezoidal)
Constant taper and sweep (trapezoidal)
Elliptic (with straight sweep of line of local a.c.)
Elliptic (with straight sweep of line of local a.c.)
Any (with straight sweep of line of local a s . )
Proportional to wing chord (uniform C,<,)
Elliptic
Elliptic (uniform
Ch,)
Elliptic
360 Appendix C. Mean Aerodynamic Chord, Mean Aerodynamic Center and Cmmcw
where A = aspect ratio, b2/S
h = taper ratio, c,/c,
c, = wingtip chord
The length of the chord through the centroid of area of a trapezoidal halfwing is equal to c. For the same wing with uniform spanwise lift distribution (i.e., C," =
const) and local aerodynamic centers on the nthchord line, the m.a. center also lies on the chord through the centroid of area. The chord through the centroid of area of a wing having an elliptic planform is not the same as T , but the m.a. center for elliptic loading and the centroid of area both lie on the same chord (see Yates, 1952).
C.3 m.a. Chord and m.a. Center for Swept and Tapered Wings (Subsonic)
The ratio F/c, is plotted against A in Fig. C.2 for straight tapered wings with stream wise tips. The spanwise position of the m.a. center of the halfwing (or the center of pressure of the additional load) for uniform spanwise loading is also given in Fig. C.2. These functions are given in Table C. 1.
The m.a. chord is located by means of the distance x of the leading edge of the m.a. chord aft of the wing apex:
b 1 1 + 2 h x =     tan A,
2 3 l + h
 1 + 2h
 12
c,A tan A,
where A, = sweepback of wing leading edge, degrees. The sweepback of the leading edge is related to the sweep of the nthchord line
A, by the relation
1  A A tan A, = A tan A, + 4n 
l + h (C.322)
Using (C.3,2) and the expression for T/c,, x can be obtained in terms of T and A, from
x (1 + 2A)(1 + A)    tan A, + 4n  2 8(1 + A + h2) l + h
The fractional distance of the m.a. center aft of the leading edge of the m.a. chord, h,_, is given for swept and tapered wings at low speeds and small incidences in Fig. C.3. The dotted lines show the aerodynamiccenter position for wings with unswept trailing edges. The curves have been obtained from theoretical and experi mental data. The curves apply only within the linear range of the curve of wing lift against pitching moment, provided that the flow is subsonic over the entire wing. The probable error of hnW given by the curves is within 3%.
Figure C.2 Mean aerodynamic chord for straight tapered wings; and spanwise position of mean aerodynamic center for uniform spanwise loading (i.e., constant C,,) (From "Notes on the Mean Aerodynamic Chord and the Mean Aerodynamic Center of a Wing" by A. H. Yates, J. Roy. Aero. Soc., June 1952.)
The total load on each section of a wing has three parts as illustrated by Fig. C.4a. The resultant of the local additional lift I,, is the lift L , acting through the m.a. center (Fig. C.4b).
The resultant of the distribution of the local basic lift 1, is a pitching couple whenever the line of aerodynamic centers is not straight and perpendicular to x. This couple is given by
362 Appendix C. Mean Aerodynamic Chord, Mean Aerodynamic Center and Cmacw
0.3
0.2
B 1
0.1
A = 1.0
0 I 0 10 20 30 40
I 50 60 70
All4. deg
Figure C.3 Chordwise position of the mean aerodynamic center of swept and tapered wings at low speeds expressed as a fraction of the mean aerodynamic chord. (From Royal Aeronautical Data Sheet Wings 08.01.0 1 .)
M e a n aeroaynarnlc CB I I L~ I
(61
Figure C.4 (a) Section total load. (b) Wing loads.
since the resultant of 1, = 0. Then
The resultant of the mu,. distribution is given by
The total pitchingmoment coefficient about the m.a. center is then
Cn,, , , ,, = Cnrl + C,n? = const (C.4,3)
If C,,,,( is constant across the span, and equals C.,,, then (C.4.2) also becomes the defining equation for C.
A P P E N D I X D
The Standard Atmosphere and Other Data
The Standard Atmosphere The tables that follow are derived from The ARDC Model Atmosphere, 1959, by Minzner, R. A., Champion, K. S. W., and Pond, H. L. Air Force Cambridge Research Center Report No. TR59267, U.S. Air Force, Bedford, MA, 1959. The values in the tables are the same, for most engineering purposes, as those derived from U.S. STAN DARD ATMOSPHERE, 1976. Prepared by the USAF, NASA, and the NOAA.
English Unitsa
Speed of Density p, sound, Zb sec2/ft4 fr/sec
Kinematic viscosity, f?/sec
Altitude Temperature h, ft T, O R
Pressure P, lb/@
21 16.2 2040.9 1967.7 1896.7 1827.7 1760.9
The Standard Atmosphere 365
English Unitsu (Continued) p~ 
Speed of Kinem~ztic Altitude Trmperuture Pressure P, Density p, sound, viscosity,
h , f i T , O R lb/ft2 lb sec2/ft4 ft/sec ft2/sec
366 Appendix D. The Standard Atmosphere and Other Data
English Unitsa (Continued)
Speed of Kinematic Altitude Temperature Pressure P, Density p, sound, viscosity,
h, f t T , O R lb@ lb sec2/ft4 ft/sec @/set
"Note: the notation xvr" means xxx X lo".
SI Units
Tempera Speed of Kinematic Altitude ture Pressure Density Sound Viscosity,
h, m T, K P ~ / m ' p kg/m3 ids m2/s
The Standard Atmosphere 367
SI Units (Continued)
Altitude h, nz
Tempera ture T, K
249.20 247.25 245.30 243.36 241.41 239.47 237.52 235.58 233.63 23 1.69 229.74 227.80 225.85 223.9 1 22 1.97 220.02 21 8.08 2 16.66 2 16.66 2 16.66 2 16.66 2 16.66 2 16.66 2 16.66 2 16.66 2 16.66 2 16.66 2 16.66 21 6.66 2 16.66 216.66 216.66 2 16.66 2 16.66 2 16.66 2 16.66 2 16.66 216.66 216.66 2 16.66 2 16.66 2 16.66 2 16.66 2 16.66
Pressure Density P ~ / m ' p kg/m3
Speed of Sound
m/s
Kinematic Viscosity,
m2/5
368 Appendix D. The Standard Atmosphere and Other Data
Other Data Conversion Factors
Multiply BY To Get
Pounds (Ib) 4.448 Newtons (N) Feet (ft) 0.3048 Meters (m) Slugs 14.59 Kilograms (kg) Slugs per cubic foot (slugs/ft3) 515.4 Kilograms per cubic meter (kg/m3) Miles per hour (mph) 0.447 1 Meters per second (mls) Knots (kt) 0.5151 Meters per second (rnls) Knots (kt) 1.152 Miles per hour (mph)
Gravity
g = 32.2 ft/s2 = 9.81 m/s2 at sea level.
A P P E N D I X E
Data for the Boeing 7471 00
The Boeing 747 is a highly successful, large, fourengined turbofan transport aircraft. The model 100 first entered service in January 1970, and since then it has continued to be developed through a series of models and special versions. As of May 1990, only versions of the model 400 were being marketed. By the year 1994, close to 800 Boeing 747s were in operation around the world, and the aircraft was still in produc tion.
The data for the Boeing 747100 contained in this appendix are based on Heffley and Jewel1 (1972). A threeview drawing of the aircraft is given in Fig. E.1. A body axis system F, is located with origin at the CG and its xaxis along the fuselage refer ence line (FRL). The CG is located at 0.25 ? (i.e., h = 0.25), and this is the location that applies for the tabulated data. The thrust line (TL) makes an angle of 2.5" with respect to the FRL as shown.
Three flight cases are documented in the data tables. They all represent straight and level steadystate flight at a fixed altitude. Case I has the aircraft in its landing configuration with 30" flaps, landing gear down, and an airspeed 20% above the stalling speed. Cases I1 and I11 represent two cruising states with the flaps retracted and the gear up.
The data in Table E. 1 define the flight conditions that apply to the three cases. It should be noted that the moments and product of inertia are given relative to the body frame F, shown in Fig. E. 1. Here the weight and inertias for Case I are smaller than those for the other two cases because the amount of fuel on board during landing is less than that during the cruise. If the data are to be applied to a reference frame dif ferent from F, (e.g., to stability axes F,) then the given inertias will have to be trans formed according to (B. 12,3). Note that F, can be rotated into F, by a single rotation of 5 about the yaxis. Values for 5 are contained in Table E. 1.
The dimensional derivatives corresponding to F, of Fig. E. 1 are contained in Ta bles E.2 to E.4. Since F, can be rotated into the stability axes F, by a single rotation of 6 about the yaxis, it follows that the transformations of (B. 12,6 and B. 12,7) can be used to obtain the derivatives corresponding to F,. Values for 8 are contained in Table E.I.
Appendix E. Data for the Boeing 747100 371
Table E. l Boeing 747100 Data
(S= 5,500ft2, b = 195.68ft,E= 27.31 ft,h = 0.25)
Table E.2 Boeing 747100 Dimensional Derivatives
Case I (M = 0.2) Longitudinal
Lateral
372 Appendix E. Data for the Boeing 747100
Lateral
Table E.3 Boeing 747100 Dimensional Derivatives
Case I1 (M = 0.5) Longitudinal
u (ftls)
w (ftls)
q (radls)
w (ft/s2)
8,
u (ftls)
p (radls)
r (radls)
8,
8, (rad)
x (lb)
4.883 X 10'
1.546 X lo'
3.994 x lo4
Table E.4 Boeing 747100 Dimensional Derivatives
Case I11 (M = 0.9) Longitudinal
y fib)
1.625 X lo3
1.342 x lo5
u (ftls)
w (ftls)
q (radls)
w (ft/s2)
8, (rad)
u (ftls)
p (radls)
r (radls)
8, (rad)
6, @ad)
z (lb)
1.342 X lo3
8.561 X lo'
 1.263 X lo5
3.104 X 10'
3.341 X lo5
M (jtdb)
8.176 X lo3
5.627 X lo4
1.394 X lo7
4.138 X lo3
3.608 X lo7
L (jtdb)
7.281 X lo4
1.180 x lo7
6.979 X 10'
2.312 X lo6
3.073 x lo6
Lateral
x (lb)
3.954 X lo2
3.144 X lo2
1.544 x lo4
y (lb)
1.198 X lo'
7.990 x lo4
N ( f t .lb)
4.404 X lo4
2.852 X 10"
7.323 X 10'
7.555 x lo5
1.958 x lo7
z (lb)
8.383 X lo2
7.928 X lo3
1.327 X lo5
1.214 X lo2
3.677 x lo5
L (frlb)
2.866 X lo4
8.357 X 10'
5.233 X 10"
3.391 x 10"
2.249 X 10"
M f f r .lb)
2.062 X lo'
6.289 X lo4
 1.327 X 10'
5.296 x lo3
4.038 x lo7
N (ftdb)
5.688 X lo4
5.864 X lo5
7.279 X 10"
4.841 x lo5
2.206 X lo7
References 373
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374 Appendix E. Data for the Boeing 747100
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to Inertia Cross Coupling. RAE Rept. No. Aero 2604, April 1958. Priestley, E. A General Treatment of Static Longitudinal Stability with Propellors, with Application
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Ribner, H. S., and Ellis, N. D. Aerodynamics of Wingslipstream Interaction. CASI Transactions, vol. 5, no. 2, 1972.
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Additional Reading
Anderson, J. D. Introduction to Flighr. 2nd ed. McGraw Hill, Inc., New York, 1985. Bertin, J. J., and Smith, M. J. Aerodynamics for Engineers. 2nd ed. Prentice Hall, NJ, 1989. Blakelock, J. H. Automatic Control of Aircraft and Missiles. 2nd ed. John Wiley & Sons, Inc., New
York, 199 1. McCormick, B. W. Aerodynumics, Aeronautics, and Flighr Mechanics. John Wiley & Sons, Inc.,
New York, 1994. Nelson, R. C. Flight Stabiliry and Automatic Control. McGraw Hill Inc., New York, 1989. Shevell, R. S. Fundamentals ofFlight. 2nd ed. Prentice Hall, NJ, 1989.
I N D E X
Accelerometer, 27 1 Adjoint matrix, 209, 304 Aeroballistic range, 6 Aerodynamic center, 24, 356 Aerodynamic transfer functions, 1 11, 129 Aeroelastic effects, 29 Aeroelastic oscillations, 5 AFCS, 7,206 Aileron, 86
adverse yaw, 87 reversal, 72, 88
Airspeed, 15 Altitude and glide path control, 275 Angle of attack, 17 Angle of sideslip, 17 Angular momentum, 95, 96 Apparent mass, 145 Approximation:
Dutch Roll mode, 195,25 1 longitudinal modes, 17 1 roll mode, 193 spiral mode, 193
Atmosphere standard, 363 Atmospheric turbulence, response to, 106,295 Automatic flight control system(AI:CS), 7,
206 Autopilot, 8 Autorotation, 15 1 Axes, 15, 101, 102
Bairstow, 3 Bernoulli, 3 Bisplinghoff, 73 Bode diagram, 217,227 Body axes, 16, 101, 102 Boeing 747, data, 368 Boost gearing, 49 Bryan, 3, 110 Buckingham's T theorem, 1 15 Buffeting, 72
Canard configuration, 23 Centre of pressure, 134 CG limits, 74 Characteristic:
determinant, 162 equation, 162 polynomial, 162, 209
Cofactor, 304 Complex amplitude, 2 15 Compressibility, 29 Computational fluid dynamics, 6 Constantpower propulsion, 69, 132 Constantthrust propulsion, 69, 132 Control, 6
of altitude, 277 closed loop, 259 derivatives, 228 equations, 206 force, 9, 12
gadgets, 64 gradient, 5 1
per g, 60 to trim, 48
lateral, 205, 207, 280 longitudinal. 33,204,207 open loop, 204 reversal, 64 roll, 86, 291 vector, 104,228 of yaw, 80
Controlfree maneuver point, 63 Controls:
displacement, 88 lateral, 207 longitudinal, 207 power, 1 1 power boost, 49 primary, 204 rate, 88
Convergence, I62 Conversion factors, 367 Convolution, 142 Convolution integral, 309 Cooper and Harper, 1 1 Crossflow, 25 Cross product, 95 Cycles to double (or half), 163
Damping: ratio, 164, 266 in roll, 150 in yaw, 154
Davies, 70 Degree of freedom, 6 Delta wing, 23
378 Index
Derivatives: aeroelastic, 156 C,,, 141, 319
CL,, 62
c,,, 34 CIp' 150' 345 C,r, 154, 349 C,,, 29, 141 c,,, 34 Cmq, 62, 135,340 C,", 131, 134 C,,, 148, 340 C"p, 15 1,345 Cnr, 154, 349 CTU, 132 C,", 131, 133 Cyp, 149 Cyr, 154 C,, 148 CL4, 135 C,", 131, 133 control, 244
jet airplane, 243 nondimensional, 207
cross, 15 1 jet transport, 166, 188 L,, 120 L,, 120 L,, 120 lateral, 156 longitudinal, 155 nondimensional, general aviation airplane,
238 quasistatic, 138 Z,, 119 Z", 118 Z,, 119 Z;, 119
det, 36 Determinant, 304 Dihedral:
lateral, 83 longitudinal, 27
Dirigible, 1 Disposable load, 74 Divergence, 7, 162 Dowell, 73 Downwash, 26,65,332
lag of, 147 Drag, 19 Dutch Roll, 13
approximation, 25 1
Dynamic gain, 2 17 Dynamic pressure, 23, 30 Dynamics of flight, 18
Earth curvature, 3 EAS, 37 Eigenvalue, 16 1. 210
computation of, 3 15 of jet transport, 166
Eigenvector, 16 1 computation of, 3 15 jet transport, 1671 69
Elastic degrees of freedom, 120 Elevator:
angle per g, 60,242 angle to trim, 35, 38 gearing, 49
Ellis, 70 Equations of motion:
Euler's, 100 general, 104 lateral, 1 13 linear, 1 11 longitudinal, 1 12 nondimensional, 1 17 state vector form, 114
Equilibrium, 6, 20 states, 18
Equivalent airspeed, 37 Etkin, 3, 141 Euler angle rates, 100 Euler angles, 98 Euler's equations of motion, 100 Extreme value theorems, 308
Feedback, 260 Feedforward, 298 Feel, synthetic, 48 Filotas, 147 Fin, 86 Fixed control, 44 Flaps, 23,64 Flax, 20 Flexible fuselage, 73, 134 Flight, 1 Flight path, 99
jet transport, 168 Flight simulator, 6 Flutter, 72 Flying wing, 22 Forced oscillation, 146 Fourier transform, inverse of, 21 1 Frame of reference, 15 Free elevator, 44
Index 379
Freetlight model, 6 Free oscillation, 146 Free rudder, 8 1 Frequency:
circular, undamped, 164 reduced, 138, 145 response, 2 14, 243
Gain margin, 265 Gates, 3 General aviation airplane, 237 Generalized inertia, 123 Generalized force, 158 Giesing, 147 Glauert, 3 Gliding flight, 132 Gravity, 3, 367 Ground effect, 74 Groundspeed, 15 Gust alleviation, 295 Gyrostabilized, 256
Handling qualities, 5, 1 I Heat transfer, 3 Heaviside theorem, 307 Henson, 23 Heppe and Celinker, 256 Highlift devices, 64 Highwing airplane, 85 Hinge moment, 4 1,324
rudder, 8 1 Hughes, 3 Human pilot, 5, 7, 8 Hunsaker, 3 Hypersonic, 20
Impulse, 207 Impulse response, 210 Induced velocity, 25 Inertia force, 124 Inertial coupling, 255 lnertia matrix, 97 Instability:
dynamic, 7 static, 7
Integral control, 267 Interference, 26 Interference effects, 25 Inverse matrix, 209 Inverse problems, 4, 107 Inver\ion theorem, 21 0
Jet: engine, 70 induced inflow, 72
normal force, 70 pitch damping of, 141
Jet transport, 1 lateral modes, 186 longitudinal modes, 165
Jones, 3
Kleinman, I I Krendel, 1 1 Kuethe and Chow, 133 Kuhn. 70
Lagrange, 3 Lagrange's equations, 122 Lanchester, 3, 172 Landing, 74 Laplace, 3 Laplace transform, 208, 304, 305
inverse of, 306 Lateral aerodynamics, 76 Lateral steady states, 237 Lawrence, 20 Leadingedge suction, 152 Lift, 19, 23 Liftcurve slope, 25, 3 1 Lilienthal, 33 Linear air reactions, 109 Linear algebra, 303 Linearization, I08 Load factor, 230, 240 Locus, transfer function, 223 Logarithmic decrement, 164 Longitudinal forces, 19 Lowwing airplane, 85
Maneuverability, 60 Maneuvers, 8 Mass ratio, 61 Mathematical model, 93 Matrix, 303
adjoint, 304 inverse, 304, 308 system, 161
McCormick, I9 McRuer, 11 Mean aerodynamic center, 356 Mean aerodynamic chord, 356 Miele, 19 Miles, 146 Millikan, 70 Modes:
effect of CG location, 182 effect of density gradient, 180 effect of speed and altitude, 177
380 Index
Modes: (continued) divergent, 162 jet transport, Dutch Roll, 190 jet transport, effect of flight condition, 176 jet transport, roll convergence, 189 jet transport, spiral, 188 lateral, approximations, 193 lateral, effect of speed and altitude on, 191 longitudinal, approximations, 17 1 longitudinal, of jet transport, 165 natural, 162 oscillatory, 162 phugoid, 166 phugoid approximation, 172 shortperiod, 166 shortperiod approximation, 174
Moment of momentum, 95 Motion, 1
lateral, 18 longitudinal, 18
Moving frame of reference, 3 16
Nacelles, 25 Neutral point, 12,29
effect of bodies on, 334 elevator free, 45 flight determination, 40
Newton, 3 Newton's laws, 93 Nichols diagram, 265 Nondimensional system, 115, 116 Nonlinear effects, 206 Normal modes, 12 1 Nyquist diagram, 2 17
Oscillating wings, 144
Parabolic polar, 68 Partial fractions, 306 Performance, 5 Period, 163 Phase margin, 265 Phillips, 256 Phugoid:
approximation, 23 1 oscillation, 166, 205
Pinsker, 256 Pitch attitude controller, 266 Pitching moment, 23
of the body and nacelles, 25 of propulsive system, 28 of the tail, 26 of the wing, 24
Pitch stiffness, 18, 21, 29
Plant identification, 4 Poles, 2 10,2 17 PrandtlGlauert rule, 133 Priestly, 70 Principal axes, 102 Propeller, 67,69
effects, 336 fin effect, 80 normal force, 80 slipstream, 70
Proportional control, 267 Propulsive system, 28
effect on pitch stiffness, 66 effect on trim, 66
Quasistatic deflections, 121 Quasisteady flow, 156
Rate control, 267 Reference flight condition, 132 Reference steady state, 108 Reid, 13 Resonance, 226 Response:
to control, 5 to elevator, 229 initial, 204 longitudinal, 228 quasistatic, 227 steady state, 204 of systems, 207 to the throttle, 235 transient, 204, 252 to turbulence, 5, 295
Reversal of slope. 39 Ribner, 79 Rigidbody equations, 93 Rodden, 147 Roll damper, 28 1 Roll stiffness, 8 1 Root locus, 267,28 1 Rotational stiffness, 76 Rotation matrix, 99 Rotors, 103 Routh's criteria, 164 Rudder forces, 9, 8 1 RungeKutta integration, 206
SAS, 7 Scalar product, 303 Schlichting, 19 Sensors, 263 Servomechanism, 260 Shock tube, 6
Index 381
Shortperiod approximation, 234 Sideslip, 237 Sideslip angle, 17, 77 Sidewash angle, 78 Sleeman, 70 Slipstream effects, 336 Slots, 64 Smalldisturbance theory, 107 Smelt, 70 Speed controller, 270 Speed stability, 177, 255 Spinning, 12 Spiral mode, 12 Spiralfroll approximation, 248 Spoilers, 87 Stability, 6
axes, 102 augmentation system(SAS), 7, 76 boundary. 75 closed loop, 264 controlfixed, 7 controlfree, 7 derivatives. 129 dynamic, 12 inherent, 7 margin, 40 of small disturbances, 5 static, 12
limit, 40 synthetic, 7 of uncontrolled motion, 16 1 weathercock, 77
Stability derivatives: dimensional, 1 18 nondimensional, 1 17
Stalling, 12 State vector, 1 14, 161 Static gain, 214 Static longitudinal stability:
general theory, 175 static margin, 29 static stability criterion, 176 static stability limit, 40
Step response, 2 12, 23 1 Stick movement, 10 STOL airplane, 70
basic data, 185 longitudinal characteristics, 184
Stringfellow, 23 Structure flexibility, 72 Submarine, 1 Sweptback wing, 23 System:
firstorder, 2 19
highorder, 209 linear invariant, 206, 207 matrix, 161 secondorder, 224 theory, 4, 5
Tab effectiveness, 330 Tabs, 47
geared, 48 servo, 48 spring. 48 trim, 47
Tail, 26 efficiency factor, 27 volume, 28
Tailless aircraft, 32, 35, 36, 46, 63 Takeoff, 74 Theodorsen function, 145 Thrust, 29
coefficient, 30, 129 line, 129 vector. 19
Time constant, 21 1 Time to double (or half), 163 Tobak, 142 Trajectories, 5 Transfer functions, 208
closed loop, 28 1,27 1 open loop, 264
Transformation: of coordinates, 3 10 of a derivative of a vector, 313 of inertias, 35 1 matrix, 3 1 1 of a matrix, 3 15 of stability derivatives, 354 of a vector, 3 10
Transient states, 8 Trim curves, 30 Trimslope criterion, 30 Trimmed lift curve slope, 36 Truckenbrodt, 19 Truitt, 20 Turbulence, atmospheric, 8 Turn, steady, 238
Unit step, 207 Unit vectors, 303 Upwash, 71 USAF. 20
Vector product, 303 Vectors, 303 Vehicle, 1
382 Index
Vertical tail, 78 volume, 79
Vibration mode, 156 Virtual displacement, 124 Vortex system, 25 Vorticity, 65
Weight coefficient, 61 Weil, 70 White noise, 207 Wind, 16
downburst, 197 effects, 196 gradient, 197, 199 turbulence, 196
Wind tunnel, 6 Wing:
bending, 157 divergence, 72 sweep, 86 wake, 26
Wright brothers, 3, 23
Yaw damper, 28 1,287 Yawing moment, 78 Yaw stiffness, 77
Zerolift line, 3 1 Zeros, 2 17