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Dynamics of Atmospheric Flight
By Bernard Etkin
Dover Publications, Inc.Copyright © 2000 Bernard Etkin
All rights reserved.
This book is about the motion of vehicles that fly in the atmosphere. As such it belongs to the branch of engineering science called applied mechanics. The three italicized words above warrant further discussion. To begin with fly—the dictionary definition 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 arrow flies," so the notion of sustention (lift) is not necessarily implied. Even the atmospheric 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 mechanical standpoint—the weight in each instance is balanced by buoyancy. They are simply 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 the general kind of system we are concerned with. Note that by the above definition a vehicle does not necessarily have to carry goods or passengers, although it usually does. The logic of the definitions is simply that the underlying engineering science is common to all these examples, 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 flying—for example, surface ships, hydrofoil craft and aircushion vehicles. In this connection it is worth noting that developments of hydrofoils and ACV's are frequently associated with the Aerospace industry. The main difference between these cases, and those of "true" flight, is that the latter is essentially three-dimensional, whereas the interface vehicles mentioned (as well as cars, trains, etc.) move approximately in a two-dimensional 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 and flying, we go on to look more carefully at what we mean by motion. It is convenient to subdivide it into several parts:
(i) Trajectory of the vehicle mass center.
(ii) "Attitude" motion, or rotations of the vehicle "as a whole."
(i) Relative motion of rotating or articulated sub-systems, such as engines, gyroscopes, or aerodynamic control surfaces.
(ii) Distortional motion of deformable structures, such as wing bending and twisting.
(iii) Liquid sloshing.
This subdivision is helpful both from the standpoint of the technical problems associated with the different motions, and of the formulation of their analysis. It is surely self-evident that studies of these motions must be central to the design and operation 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. 1.1. 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 restricted set of all possible cases. Its emphasis is on the flight of airplanes in the atmosphere. The general equations derived, and the methods of solution presented, are however readily modified and extended to treat the other situations that are embraced by the general problem.
All the fundamental science and mathematics needed to develop this subject existed 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, provided 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 practice. This body of knowledge is well documented in several texts of that period, e.g. ref. 1.4. Concurrently, principally in the USA and Britain, a large body of aerodynamic data was accumulated, 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 quantum-dynamical processes, including all of "ordinary" and much of celestial mechanics. What then distinguishes flight dynamics from other branches of applied mechanics? Primarily it is the special nature of the force fields with which we have to be concerned, the absence of the kinematical 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:
(v) Solar radiation
We should observe that two of these fields, aerodynamic and solar radiation, produce important heat transfer to the vehicle in addition to momentum transfer (force). Sometimes we cannot separate the thermal and mechanical problems (ref. 1.5). 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 remarked that even in atmospheric flight the gravity force can not always be approximated as a constant vector in an inertial frame. Rotations associated with Earth curvature, and the inverse square law, become important in certain cases of high-speed and high-altitude flight (Chapters 5 and 9).
The prediction and measurement of aerodynamic forces is the principal distinguishing feature of flight dynamics. The size of this task is illustrated 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) and of compound geometries (wing + body + tail).
As remarked above, Newton's laws state the connection between force and motion. 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 kind—the system and the motion are given and the forces have to be calculated.
2. Inverse problem of the second kind—the forces and the motion are given and the system constants have to be found.
3. Mixed problems—the 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 that is of great current interest in system theory. (Inverse problems were treated in Chapter 11 of Dynamics of Flight—Stability and Control, but are omitted here.)
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 human-pilot/machine 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.
THE TOOLS OF FLIGHT DYNAMICISTS
The tools used by flight dynamicists to solve the design and operational problems of vehicles may be grouped under three headings:
The analytical tools are essentially the same as those used in other branches of mechanics. Applied mathematics is the analyst's handmaiden (and sometimes proves to be such a charmer that she seduces him away from flight dynamics). One important branch of applied mathematics is what is now known as system theory, including stochastic processes and optimization. It has become a central tool for analysts. Another aspect of this subject that has received a great deal of attention in recent years is stability theory, sparked by the rediscovery in the English-speaking world of the 19th century work of Lyapunov. At least insofar as manned flight vehicles are concerned, vehicle stability per se is not as important as one might suppose. It 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 handling qualities. It isperformance criteria that really matter, so to expend a great deal of analytical and computational effort on finding stability boundaries of nonlinear and time-varying systems may not be really worthwhile. On the other hand, the computation of stability of small disturbances from a steady state, i.e. the linear eigenvalue problem that is normally part of the system study, is very useful indeed, 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 machine computation has revolutionized practice in this subject over the past ten years. Problems of system performance, system design, and optimization that could not have been tackled at all a dozen years ago are now handled on a more or less routine 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 rocket-boosted and gun-launched free-flight model techniques. Hand in hand with the development of these general facilities has gone that of a myriad of sensors and instruments, mainly electronic, for measuring forces, pressures, temperatures, acceleration, angular velocity, etc.
Second, we must mention the flight simulator as an experimental tool used directly by the flight dynamicist. In it he studies mainly the matching of the man to the machine. This is an essential step for radically new flight situations, e.g. space capsule reentry, or transition of a tilt-wing VTOL airplane from hovering to forward speed. The ability of the pilot to control the vehicle must be assured long before the prototype stage. This cannot yet be done without test, although limited progress in this direction is being made through studies of mathematical models of human pilots. The prewar Link trainer, a rudimentary device, has evolved today into a highly complex, highly sophisticated apparatus. Special simulators, built for most new major aircraft types, provide both efficient means for pilot training, and a research tool for studying flying qualities of vehicles and dynamics of human pilots.CHAPTER 2
This chapter contains a summary of the principal analytical tools that are used in the formulation and solution of problems of flight mechanics. Much of the content will be familiar to readers with a strong mathematical background, and they should make short work of it.
The topics treated are vector/matrix algebra, Laplace and Fourier transforms, random process theory, and machine computation. This selection is a reflection of current needs in research and industry. The vector/matrix formalism has been adopted as a principal mathematical tool because it provides a single powerful framework that serves for all of kinematics, dynamics, and system theory, and because it is at the same time a most suitable way of organizing analysis for digital computation. The treatment is intended to be of an expository and summary nature, rather than rigorous, although some derivations are included. The student who wishes to pursue any of the topics in greater detail should consult the bibliography.
2.2 VECTOR/MATRIX ALGEBRA
As has already been remarked, this book is written largely in the language of matrix algebra. Since this subject is now so well covered in undergraduate mathematics courses and in numerous text books, (2.1, 2.11) we make only a few observations here.
Excerpted from Dynamics of Atmospheric Flight by Bernard Etkin. Copyright © 2000 Bernard Etkin. Excerpted by permission of Dover Publications, Inc..
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