Stability and Control of Airplanes and Helicopters

511 Pages·1964·25.72 MB·English

by  Edward Seckel (Auth.)

STABILITY
AND CONTROL
OF AIRPLANES
AND HELICOPTERS
By Edward Seckel
DEPARTMENT OF AERONAUTICAL ENGINEERING
THE JAMES FORRESTAL RESEARCH CENTER
SCHOOL OF ENGINEERING AND APPLIED SCIENCE
PRINCETON UNIVERSITY
PRINCETON, NEW JERSEY
@
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Preface
This book is about aircraft flying qualities, which determine the suitability
of the machine for control by a human pilot. These have classically been the
stability and response of the aircraft, but in recent years the operation of the
man-machine combination as a closed-loop control system has been analyzed
in some detail.
I have tried to present the former classical aircraft characteristics in adequate
analytical fashion, but always in the framework of a pilot's viewpoint. The latter
is a physical picture, not a mathematical one, and so in discussion of analytical
results, I have dwelt on physical interpretation and meaning for the pilot.
I think this matter has been a little neglected in other books, and I hope it is
a feature of this one.
The man-machine system analysis has been put forward in recent years
with great success. It is a matter of considerable interest to me and a technique
of which I heartily approve. But thorough analytical treatment of it is beyond
this book, and I have compromised with an introduction to the subject—again
with emphasis on the physical aspects and the pilot's point of view.
The treatment of rotary and fixed wing aircraft in the same book is a bit
unconventional. For the pilot, however, the two types differ only in detail.
In general, the control problem is the same one; both types of aircraft obey
the same equations; and the pilot is the same: therefore most of what is known
about one case should be applicable to the other. This has been almost ignored
in the past and developments in stability and control for the two types have
followed almost independent courses. I hope the treatment here helps to put
them on the same course, in the same perspective.
I should be pleased if this book proves useful for reference, and I think
parts of it may be. But I most hope for its success as a text. It begins with
material on static stability and control, usually taught to college juniors. Its
depth and detail, however, are more appropriate to graduate school coverage
of the subject. The same is true of the dynamic stability chapters: the mathe-
matical level is not beyond junior or senior undergraduates in modern engineering
curricula, but the detail and scope are typical of graduate courses. The material
on helicopters is intrinsically no more difficult, but there is usually no place
in undergraduate curricula for it, so it is ordinarily offered at graduate level.
I therefore hope the book will be useful for teaching from introduction at
undergraduate level through first year graduate work. In any case, I would
expect the instructor in a particular course to select sections appropriate to
his level, and not necessarily to go straight through.
v
VI PREFACE
Although it would not be essential to supplement the helicopter section with
other reading, it would certainly be helpful to do so, especially about basic
rotor aerodynamics and introductory concepts of design, configuration, per-
formance and control. The treatment of rotor aerodynamic forces is rather
handbookish, because, in fact, it is taken almost directly from a U. S. Navy
Handbook on Helicopter Stability and Control· I am very much indebted to the
Bureau of Weapons, U. S. Navy Department, for permission to use the material.
The problems throughout are based on real aircraft, selected to represent
the gamut from simple to complicated, and from conventional utility designs
to futuristic research types. But they are real aircraft, and the data given are
real data. Many of the problems involve comparison of theory and experiment
to demonstrate their mutual relationship. The student will see that rarely do
they agree perfectly. I expect him to sustain this revelation, and eventually to
agree that the fact helps make engineering interesting and challenging. Some
of the problems, consisting of several parts for many airplanes, are obviously
too long, and the parts for assignment must be carefully selected by instructors.
Some are better suited to laboratory experiments or even term projects. Many
involve extensive laboratory equipment like airplanes, instrumentation, wind
tunnels, models, analog computers, etc. I hope these problems will encourage
the acquisition of such facilities and their use in teaching this subject. I cannot
see how to teach it properly without some such exercises.
I am very much indebted to the following individuals and organizations for
supplying and reviewing the data for problems, as follows:
The Grumman "Mohawk":—S. W. Rogalski, Aerodynamics, Grumman
Aircraft Engineering Corporation, Bethpage, Long Island, New York.
F-100 and X-15:—H. H. Crotsley, North American Aviation, Inc., Los
Angeles, California.
X-l and X-15:—P. Bikle, NASA Flight Test Center, Edwards Air Force
Base, Edwards, California.
B-58 Hustler:—F. A. Curtis, Convair Division, General Dynamics
Corporation, Fort Worth, Texas.
Boeing 707:—R. J. White, Transport Division, Boeing Airplane Company,
Renton, Washington.
S-58 Helicopter:—D. Cooper, Sikorsky Helicopter Company, Bridgeport,
Connecticut.
They have contributed greatly to the project and I am very grateful for their
time and interest.
The fixed-wing section of the manuscript has been read by R. J. White
of Boeing; and the helicopter part by H. C. Curtiss, Jr., of Princeton University,
and Dean Cooper of Sikorsky. I appreciate very much their important criticism
and their effort, which was considerable.
PREFACE VÜ
I am indebted to H. C. Curtiss, Jr. for much more than just his review.
Much of the treatment of helicopter dynamics is based on his published work,
and many of my concepts and ideas in this area evolved out of discussions
with him. I am quite happy to acknowledge his substantial part in the helicopter
section.
The preparation of this book was undertaken during sabbatical leave from
teaching duties at Princeton University. This support, and the encouragement
and help of Professor Courtland D. Perkins, Chairman, Aeronautical Engineering
Department, were essential. I offer him my earnest appreciation.
For the students, on whom I have practiced this material through the years,
I have sympathy, and thanks for their patience, interest, and help.
Finally, I offer my thanks to Mrs. Grace Arnesen for her outstanding devotion
and good humor in typing the manuscript.
EDWARD SËCKEL
To Debbie and Dave
CHAPTER I
Some Aerodynamic Generalities
For the purposes of estimating stability, control response, and handling
qualities, the analyst needs to know the lift and moment characteristics
of the airplane, which is usually broken down for purposes of estimating,
into component parts. The parts naturally divide into three groups:
wings and associated accessories; bodies, like fuselages, nacelles, tip tanks,
and so forth; and control surfaces. We consider here, in very general
terms, certain principles about the aerodynamics of these things.
1. Wings of Medium to High Aspect Ratio
Distribution of Lift
The lift on a wing is usually described by the distribution over its
surface of the aerodynamic loading. The loading, or load per unit area,
is the difference between upper and lower surface pressures at a point in
the plan view. The distribution of loading can be shown by sketches like
Fig. 1. The ordinates are nondimensional coefficients of pressure, or
loading per unit dynamic pressure, C = C — C . For stability and
P P P
control estimates, two features are of interest—the total load (or lift) and
its distribution, which determines moment.
Any one of the diagrams showing the distribution of loading, at a
particular spanwise station, shows the "chordwise loading distribution ,,
for that section. The areas of these chordwise distributions represent
points on the "spanwise distribution,, diagram of load per unit span.
The area of the latter is, of course, the total lift of the wing.
The integrals of the chordwise distributions, for lift and moment
computations, are section lift and moment coefficients, c and c .
x m
They may be said to represent the lift and moment (per unit dynamic
pressure) per unit span, per unit chord, at any station. If a wing with
a given plan form has known distributions of c and c along the
x m
span, then the total load and moment follow, respectively, from the
1
2 I. SOME AERODYNAMIC GENERALITIES
simple integrations
(la)
(lb)
Basic and Additional Distributions
It is further convenient to break down these distributions into two
parts: one without net lift, indicated always by the phrase for zero lift or
the term basic, or the subscript ( ) ; and another additional part contri-
b
buting all the lift. The * 'basic' ' components of these distributions depend
on camber and twist, but they never add up to any total lift. They can
cause moments, however. The flat wing, without camber or twist, has
no zero-lift or basic components.
The additional part, contributing the lift, varies with angle of attack
but is usually assumed independent of camber or twist. In many cases it
can be assumed that the shape of this part is preserved, and only the
magnitude varies with a. In that case, the aerodynamic center is easily
defined: it is simply the centroid of the "additional" distribution; the
free moment for the "basic" distribution is identical to the whole moment
about the aerodynamic center. The whole lift and moment are obtained
by superposition of the "basic" and "additional" parts. These concepts
of independence of the two distributions, and their superposition, are
only valid and useful for wings of reasonably high aspect ratio.
We shall begin with a discussion of slope-of-the-lift-curve, which
involves only the "additional" distributions. The "basic" components
will be considered thereafter.
Slope-of-the-Lift-Curve
A study of wings always starts with the two-dimensional, or infinite
aspect ratio, case where all sections are identical. First consider a thin,
unswept airfoil. At low subsonic speeds, the lift per unit span is
(2)
This lift is distributed across the chord as shown in Fig. la, with
center at the ^-chord point, which is the aerodynamic center.
The swept two-dimensional wing has a slope-of-the-lift-curve, c ,
l
(2a)
1. WINGS OF MEDIUM TO HIGH ASPECT RATIO 3
This is predicted by resolving the oncoming flow into three com-
ponents—one normal to the plane of the wing, one spanwise and one
chordwise in the plane of the airfoil. The spanwise component has only a
viscous effect and is neglected, whereas the lift due to the others is
calculated from the usual theory without sweep. The final result is,
of course, the formula given.
Wing lift
coefficient C
Section lift L
coefficient c
Pressure coefficient
Cpl - CPU
Subsonic flow
(a)
Total force
Spanwise distribution
Chordwise distribution
Supersonic flow
(b)
FIG. 1. Aerodynamic load distribution on rectangular wings.
4 I. SOME AERODYNAMIC GENERALITIES
In the range of high aspect ratio, say above five, the Prandtl lifting-line
theory suffices pretty well to calculate the distribution of lift along the
span. In this theory the wing is replaced by a system of "bound" vortexes,
concentrated along the ^-chord line of the wing; and a trailing vortex
sheet is distributed in accord with the variation of bound vorticity along
the span and with the rule that the vortex filaments cannot end in the
fluid. The lift, and hence the bound vorticity, at any station is calculated
on the assumption that the only interference between sections is the
downwash induced by the trailing vortex sheet, which reduces the local
angle of attack. This is all pretty complicated except in the case of
elliptical plan form, where it turns out that the induced angle is constant
along the span, and so also is the section lift coefficient which equals
the wing lift coefficient. The induced angle is
* = 7Ä (3)
The result is well known:
a = i + KM) (3a)
or, for a = 2π:
0
α = 2πΑΤ2 (3b)
The A/(A + 2) has come to be used as a three-dimensional correction
factor for C . It is used also for wings of other plan form and works
L
well in the range of high A. It does not work well in the range of low
aspect ratio, where it incorrectly indicates, for vanishing A,
£ - ** (3c)
Low aspect ratio wing theory {E5.4)> which may be viewed as an
extension of slender body theory, gives a value just half this, or
± = ^- (4)
A 2 v '
An empirical formula which fits the latter as A —► 0 and also the
former for A —> oo is
πΛ (4a)
1 + Vl + Μ/βο)1
This may even be used for wings of moderate sweep, by substituting
a = 2π cos Λ,
0
(4b)
1+Λ/1 + (2^Τ)2