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PUBLISHED BY
JOHN WILEY & SONS, Inc., NEW YORK
CHAPMAN & HALL, Limited, LONDON
MATHEMATICAL MONOGRAPHS
EDITED BY
MANSFIELD MERRIMAN AND ROBERT S. WOODWARD
No. 21
THE DYNAMICS OF THE
AIRPLANE
BY
KENNETH P. WILLIAMS, PH.D.
ASSOCIATE PROFESSOR OF MATHEMATICS
INDIANA UNIVERSITY
NEW YORK
JOHN WILEY & SONS, INC.
LONDON: CHAPMAN & HALL, LIMITED
1921
W 5"
Library
COPYRIGHT, 1921
BY
K. P. WILLIAMS
PRESS Of
RRAUNWORTH li CO.
BOOK MANUFACTURERS
UROOKUYN. N. Y.
PREFACE
IT was the good fortune of the author to attend the University
of Paris during the spring semester of 1919. One of the special
courses which the French authorities, with their characteristic
hospitality, arranged for the large number of students from the
American army, was a course in aerodynamics, given by Professor
Marchis. The comprehensive knowledge that Professor Marchis
possessed of all branches of the new science of aeronautics, the
inestimable value of his advice to the French Republic during
the war, the interest he took in his rather unusual class, could not
fail to be an inspiration.
This book is an outgrowth of those parts of Professor Marchis'
lectures that were of particular interest to the author. It is in
no sense a complete treatise on aviation. Questions of design
and construction are passed over with bare mention. The book
is intended for students of mathematics and physics who are
attracted by the dynamical aspect of aviation. The problems
presented by the motion of an airplane are novel and fasci-
nating. They vary from the most pleasing simplicity to the
most stimulating difficulty. The question of stability, partic-
ularly, exhibits at the same time the elegance and the power
of analysis, and shows the adaptability of some of the general
developments in dynamics. The field is assuredly a fruitful
one of study, and increasing demands will be put upon the
mathematician as the science of aviation continues its rapid
development. The mathematician can well own a sense of pride
that he had already at hand, in the developments inaugurated
by Euler and Routh, a means of dealing accurately with the
question of stability, that plays so fundamental a role in the
science of flying.
iii
447195
IV PREFACE
The treatment in the text is for the most part elementary.
The last chapter alone demands of the student familiarity with
more advanced dynamical methods. In the treatment of descent
a slight digression is made to consider in part the nature of the
solution of a system of two differential equations. This was done
in order not to completely evade what seems a problem of con-
siderable difficulty. It might seem that a treatment of the
propeller should not find a place in a book with the purpose of
this one. No student of mathematics, however, could fail to
own a curiosity as to a propeller's action, and it is hoped the dis-
cussion, while not complete, will at least serve as a sufficient
introduction.
The various curves in the text were plotted by Mr. R. W.
Smith, a former student in this university. The author is
further indebted to the Smithsonian Institution for permission
to use Figs. 12 and 49.
In addition to the various books that are referred to in the
text the author has made use of his notes of the lectures of Pro-
fessor Marchis, translated into English by Madame Ciolkowska,
who rendered most valuable aid as an interpreter for those who
understood and spoke the language of Professor Marchis only
with difficulty.
K. P. WILLIAMS.
Indiana University,
July, 1920.
CONTENTS
CHAPTER I. THE PLANE AND CAMBERED SURFACE
ARTICLE PAGE
i, 2. PRELIMINARY CONSIDERATIONS. . ; 1-3
3. PRESSURE ON A PLANE 4
4. THE INCLINED PLANE 5
5. THE CENTER OF PRESSURE 6
6. ASPECT RATIO 6
7. THE CAMBERED WING 7
8. CHARACTERISTIC CURVES FOR A GIVEN WING 8, 9
9. POLAR DIAGRAM 10
10. EFFECT OF VARIATION OF WING. ELEMENTS n
11. PRESSURE OVER THE WING n
12. CENTER OF PRESSURE 12
13. RELATION BETWEEN K y AND K z 13
14. THE BIPLANE 14
15. BODY RESISTANCE 15
16. EXPERIMENTS ON COMPLETE MODEL 15
CHAPTER II. STRAIGHT HORIZONTAL FLIGHT
17. PRELIMINARY CONSIDERATIONS 18
18. HORIZONTAL FLIGHT 19
19. THE VELOCITY 20
20. LANDING SPEED 20
21. EFFECT OF ALTITUDE 21
22. THE EFFORT OF TRACTION 22
23. OPT MUM ANGLE 22
24. FINENESS 23
25. OPTIMUM ANGLE, CONTINUED 24
26, 27. USEFUL POWER 25
28, 29. ECONOMICAL ANGLE 26-28
30. GENERAL CONSIDERATIONS 29
CHAPTER III. DESCENT AND ASCENT
i. DESCENT
31. PRELIMINARY CONSIDERATIONS CONCERNING DESCENT 31
32. EQUATIONS OF MOTION 31
v
VI CONTENTS
ARTICLE PAGE
33, 34. RECTILINEAR DESCENT 33~35
35. GENERAL RESULTS CONCERNING DESCENT 36
36-38. DESCENT CONSIDERING AIR DENSITY CONSTANT 38-42
2. ASCENT
39. PRELIMINARY CONSIDERATIONS CONCERNING ASCENT 43
40. EQUATIONS OF MOTION FOR ASCENT 43
41. VELOCITY ALONG PATH 44
42. THE FORCE OF TRACTION 44
43. THE POWER NECESSARY 46
44. THE VERTICAL VELOCITY 47
45. GENERAL CONSIDERATIONS 48
46. EXPERIMENTAL LAW OF VERTICAL VELOCITY 49
47. TIME OF ASCENT 50
48. DETERMINATION OF THE CEILING 50
CHAPTER IV. CIRCULAR FLIGHT
1. HORIZONTAL TURNS,
49. GENERAL CONSIDERATIONS 52
50. EQUATIONS OF MOTION 53
51. VELOCITY AND INCLINATION 54
52. THE TRACTION AND POWER 55
53. ACTION OF CONTROLS 56
2. CIRCULAR DESCENT
54. GENERAL CONSIDERATIONS 58
55. EQUATIONS OF MOTION 59
56. RELATIONS BETWEEN ANGLES USED 60
57. OTHER FORM OF EQUATIONS OF MOTION 61
58. IDENTITY OF Two FORMS OF EQUATIONS OF MOTION 62
59. DETERMINATION OF ANGLES SPECIFYING MOTION 63-65
60. THE VELOCITY 66
CHAPTER V. THE PROPELLER
61-64. GENERAL CONSIDERATIONS 69, 70
65. GEOMETRICAL PITCH 71
66-68. THE THRUST AND POWER ... 74-76
69. EFFICIENCY 77
70. EFFECT OF ALTITUDE 77
71, 72. GRAPHS OF PROPELLER COEFFICIENTS 78
73. MOTOR DIAGRAM 8 1
74-78. ADAPTATION OF MOTOR- PROPELLER GROUP TO MACHINE 82-85
CONTENTS Vll
CHAPTER VI. PERFORMANCE
i. CEILING
ARTICLE PAGE
79. GENERAL CONSIDERATIONS 86
80. DETERMINATION OF CEILING 86
81. SUPERCHARGE 89
2. RADIUS OF ACTION
82. DETERMINATION OF DISTANCE A MACHINE CAN FLY 90
83. PROBLEM OF RETURN JOURNEY 93
CHAPTER VII. STABILITY AND CONTROLLABILITY
84-88. PRELIMINARY CONSIDERATIONS 94-96
89. STATIC AND DYNAMIC STABILITY 96
90. PITCHING, ROLLING, AND YAWING 97
LONGITUDINAL STABILITY
91. PRELIMINARY CONSIDERATIONS 97
92. METACENTRIC CURVE 98
93. THE TAIL PLANE 99
94. THE ELEVATOR 100
95. GENERAL CONSIDERATIONS 102
STABILITY IN ROLLING
96. PRELIMINARY CONSIDERATIONS 102
97. DIHEDRAL 103
98. CONTROLLABILITY, AILERONS 104
LATERAL STABILITY
99. FIN, RUDDER 194
100. ACTION OF RUDDER 105
101. CONNECTION BETWEEN YAWING AND ROLLING 105
102. SPIRAL INSTABILITY . 106
CHAPTER VIII. STABILITY (CONTINUED)
103. METHOD OF BRYAN 107
104, 105. MOVING AXES 108
106. ANGULAR VELOCITIES AND MOMENTA 109
107, 108. ORIENTATION 110-112
109. EQUATIONS OF MOTIONS , 113
no. STEADY MOTION 114
Vlll CONTENTS
ARTICLE PACK
in, 112. SYMMETRIC AND ASYMMETRIC OSCILLATIONS 115-117
113. THE FUNDAMENTAL QUARTIC EQUATIONS 118
114. THE CONDITIONS OF STABILITY 120
115. THE FORM OF THE COEFFICIENTS 120
116. THE DETERMINATION OF THE COEFFICIENTS 121
117. EXAMPLE 122
118. DEPENDENCE OF STABILITY ON SPEED 123
119. FACTORED FORM OF QUARTIC EQUATIONS 123
120. EXAMPLE 1 24
121. EFFECT OF GUSTS. GENERAL CONSIDERATIONS 125
122. EQUATIONS FOR TREATING GUSTS 126
123, 124. EXAMPLE 128, 129
APPENDIX
1-3. TRANSFORMATION OF UNITS 131-133
4. TABLE OF AIR PRESSURE AND DENSITY 134
5. TABLE FOR VELOCITY TRANSFORMATION 135
6. REFERENCES 135
THE DYNAMICS OF THE AIRPLANE
CHAPTER I
THE PLANE AND CAMBERED SURFACE
1. THE possibility of aerial navigation depends upon the
solution of two problems, the problem of sustentation and
the problem of propulsion. At the very outset two distinct
courses are therefore open. We can look upon the problems
as entirely separated from each other, or we can regard them
as essentially connected.
In the first case we look for separate solutions, solving first
the problem of sustentation, and then, with this successfully
disposed of, search for a means of propulsion. This course
is historically the older and it is the simpler, for it meets
the difficulties one at a time.* A balloon filled with a light
*Early literature abounds with mythical accounts of the flights of legendary
heroes equipped birdlike with wings. Among those who seriously studied the
question of flight, and actually designed machines with wings to be attached to a
person and driven by his own muscular power, Leonardo da Vinci, the renowned
artist, occupies the first place. The first instance of people actually ascending
from the earth took place November 21, 1783, at Paris. The apparatus was a
balloon constructed by Stephen Montgolfier, and the flight covered five miles.
The construction of crude dirigibles followed within a few months. The first
instance of partial sustentation without the use of gas occurred at this same period.
On December 26, 1784, Sebastian Lenormand descended from the tower of the
Montpelier Observatory by means of two small parachutes, the idea of the para-
chute being due to da Vinci. Successful efforts with gliders were made in the last
years of the igth century. Modern aviation dates from 1903, when the Wright
brothers first constructed a machine, equipped with engines, which could actually
rise from a level field, without the assistance of air currents, and make flights con-
trolled by the pilot. For the history of aeronautics see Albert F. Zahm, "Aerial
Navigation," D. Appleton, & Company, New York and London, 1911.
2 ' ftiE DYNAMICS -O> THE AIRPLANE
gas, such as hydtogeri ^ or 4tefttinV affords -a means of susten-
tation. But aerial navigation means far more than the ability
to stay aloft, and a craft which can travel only as the wind
blows it, can serve few purposes other than that of furnishing
amusement. The question of equipping a balloon with engines
and a means of propulsion, of traveling in a desired direction
with a velocity within our control, of maintaining a desired
altitude, of rising and landing, must be answered before we
can say the balloon has furnished a means of aerial navigation.
It is only since the development of the gasoline motor that this
has been possible on any extensive scale.
The second method of attacking the problem seeks to solve
simultaneously the problems of sustentation and propulsion.
The possibility of propulsion must now come first, for susten-
tation will be obtained from the motion. It is then evident
that this method, even more than that of the balloon, had to
await the perfection of a source of energy such as the gasoline
engine. It is only with aircraft of this second sort that we are
concerned. Such a machine is called an airplane, or aeroplane.
We shall adopt the first term. Both names are suggested by
the fundamental role played by surfaces approximately plane
with which the machine is provided. The air reaction on these
surfaces, produced by the motion of the machine, furnishes the
sustentation. The complete machine will consist of other mem-
bers, and we can divide it into five distinct parts: the sustain-
ing surfaces, the stabilizing and controlling surfaces, the motor-
propeller group, the body, or fuselage, with its place for pilot,
passengers and freight, and the landing gear. To these main
parts must, of course, be added the various elements of con-
struction by which the different parts are united and the
requisite strength given to the complete machine.
We shall not give a discussion of the complete construction
of an airplane, but limit ourselves to those features which
are necessary for a comprehension of the dynamical problems
which we shall study.
2. The principles that govern the construction of an air-
plane, the phenomena that operate during its flight and deter-
THE PLANE AND CAMBERED SURFACE 3
mine its behavior, are derived from an understanding of the
laws concerning the effect of the wind upon flat and curved
surfaces. We can try to determine these laws in two ways,
mathematically or experimentally.
In the mathematical method we begin with the principles
and equations of hydromechanics. We then see if we can cal-
culate the pressure that a current of air, moving with a certain
velocity, will exert upon, for instance, a rectangular plane
surface. We must be able to do this for different inclinations
of the plane to the air stream. The problem is one of great
complexity. In order to construct differential equations that
will exhibit the phenomena, and in order to integrate these,
we must make assumptions that lead our results to differ from
carefully measured observations. For instance, we may
assume that the air is a perfect fluid, that is, that it is neither
viscous nor compressible. The last assumption seems to be
justified for the range of velocities occurring in aeronautical
work, but the assumption as to the viscosity vitiates our
results when we apply them to actual problems. Even with
the assumption of an ideal fluid it is difficult to handle the
equations involved. While we can in certain instances obtain
information as to how the air streams around obstructing
objects, and how it behaves in their vicinity, the results are
not such as to make this a simple or satisfactory method of
attacking the problem.*
The laws concerning air reaction are determined experi-
mentally in a wind tunnel, f A current of air of several feet
thickness is obtained by means of a large fan. Various and
accurately known velocities can be given to the air stream.
The surfaces upon which we wish to study the pressure are,
of course, of limited dimensions, but are similar to those which
* For an elementary mathematical treatment see Cowley and Evans, "Aero-
nautics in Theory and Experiment," Longmans, 1918, Chapter III.
f Among the various aerodynamical laboratories, can be mentioned those
of M. Eiffel at Paris, the National Physical Laboratory of England, the Massa-
chusetts Institute of Technology, and Leland Stanford University. A description
of the equipment of such a laboratory can be found in Smithsonian Miscellaneous
Collection, Vol. 62.
THE DYNAMICS OF THE AIRPLANE
are to be employed in practice. They are held in the current
of air by arms that are connected with balances constructed
so as to allow a determination of the magnitude and direction
of the air reaction. By changing the shape of the surfaces,
the velocity of the air stream, the orientation of the body, etc.,
the laws which furnish the basis for the design of an airplane,
as well as the knowledge of its behavior, are determined.
Questions relating to the stability of a machine, to the proper-
ties and efficiency of propellers, are also investigated in the
same way.
We pass to the consideration of some of the basic aerodynamic
laws, as determined empirically.
3. Pressure on a Plane. When a portion of a plane
surface is introduced normal to an air current, the pressure
produced upon it is a force that tends to displace it, and we
find by experiment that the force may be written
F = KAV 2 ,
where F is the force, A the area, V the velocity of the air
stream, and K a constant for surfaces geometrically similar.*
Strictly speaking, K depends upon the size as well as shape,
but changes slowly and seems to approach a limiting value as
the plate becomes larger. For instance, for a square plate
or circle we find, provided A is expressed in square feet, V in
miles per hour, F in pounds:
Side of Square or
Diameter of Circle
K
in Feet.
o-5
I.O
.00269
.00286
2.0
.00314
3-o
.00322
5-o
.00327
IO.O
.00327
"This law can also be obtained from theoretical considerations. Consider the
air as composed of separate particles, moving parallel and striking the plate, nor-
mal to their direction of motion. Assume that all particles are brought to rest
THE PLANE AND CAMBERED SURFACE 5
We note the slight change produced in K when the side of
the square is changed from 3 to 10, although the area is increased
over ten times.
The value of K will depend upon the units in which we
are expressing F, A , and V. It is necessary to be able to change
from one system to another. This question is discussed in the
appendix. We shall assume, unless the contrary is stated,
that forces are measured in pounds, areas in square feet, and
velocities in miles per hour.
The value of K also depends upon the density of the air.
The values given above are for a temperature of o C., and a
pressure of 760 mm. of mercury.
4. The Inclined Plane. Let the plane be inclined at
angle to the direction of air flow. This angle is called the
angle of attack. Neglecting
what is called skin friction, it
follows that the resultant force
is normal to the plane. Its
magnitude and point of applica-
tion vary with 0.* The nature
of the air flow about the plane FlG T
is quite complicated, but can be
investigated by photography. By such means information
is obtained as to how the air divides in front of the plane,
flows over the upper and lower edges, and unites again behind
the plane.
by the impact. The pressure on the plate will equal the momentum lost by the
air. The quantity of air that comes into contact with the plate in a unit of time
is pA V, where p is the density of the air. As all particles lose their velocity V,
the momentum lost will be pA V 2 . The errors in the hypothesis are apparent, but
experiment, while giving a value of the constant different from that deduced by
the reasoning above, confirms the qualitative nature of the law.
* If we denote by Fe the value of the force for the angle 6 so that F^ repre-
sents the force for the normal plane, various formulae exist for obtaining FQ.
Newton gave from theoretical reasons F0 = F 90 sin 2 6, which is totally discordant
with experiments. The formula of Colonel Duchemin is much more accurate.
He gave
2 sin 6
THE DYNAMICS OF THE AIRPLANE
We are especially interested in the vertical and horizontal
components of the total force F. We call these components
the Lift, L, and Drag, D. It is proved by experiments that
we may write
L = K V AV 2 , D = K X AV 2 ,
where K v and K x are constant for a given angle of attack.
For a square plane we can take the following values:
Angle.
Ky
Kx
5
.00045
.00007
10
.00097
.OOOI9
20
.00208
. 00074
30
.00291
.00173
5. The Center of Pressure. It is important to know the
manner in which the point of application of the resultant
pressure varies as the angle of
attack changes. The general re-
sult can be stated as follows:
As the angle of attack diminishes
the center of pressure approaches
the leading edge.
This behavior of the center of
pressure is shown in Fig. 3.
A knowledge of the movement
of the center of pressure is of
importance in the subject of sta-
30
90 c
FIG. 2.
bility and in finding the forces on the
control surfaces.
6. Aspect Ratio. It was stated
above that the quantity K in the
fundamental equation for the pressure
was constant for planes geometrically
similar. In case we have a rectangle
of sides a and b, situated, as shown,
in an air current, we call the fraction
FIG. 3.
a/b the aspect ratio. This quantity is of importance. We
find that the coefficients K, K v , K x , which have been used
THE PLANE AND CAMBERED SURFACE
above, vary when the aspect ratio is changed. Those values
which have been given for a square may, however, be used
with fairly accurate results for planes with aspect ratio close
to unity. The dependence of the quantities K, K V) K x on the
aspect ratio comes from the important effect of the boundary
of the surface upon the resistance. If we have a long narrow
rectangle, it is evident that the escape of the air around the
boundary will greatly alter the pressure from what it would
be for the equivalent square plane.
7. The Cambered Wing. If we examine a bird wing, we
find that it is not flat, but is curved. This suggests that there
may be some aerodynamic advantage in such a surface. Experi-
ment amply confirms this, and the sustaining surfaces of all
airplanes are curved, or cambered. Evidently also the surface
must have thickness for constructional reasons.
The word aerofoil is used to designate a sustaining surface
or wing. In Fig. 4 there is illus-
trated the general shape of the
section of an ordinary aerofoil. We
call AB the chord, DC the camber
of the upper surface, EC the camber
of the lower surface, C the position of maximum camber. The
quantities BC, CD, CE are generally expressed in terms of
the chord AB.
Let the wing be placed with reference to the air flow as
shown in Fig. 5. Then by the angle
of attack is meant the angle be-
tween the chord and the direction
of the relative wind. Let N repre-
sent the direction of the normal and
R the resultant pressure. It is found
that for small angles R is in advance
of N. This increases the lift L, and
diminishes the drag D. These are
desirable effects, and it is partly on account of this property that
the cambered wing is more efficient for sustaining purposes
than the plane wing.
FIG. 4.
FIG. 5.
8
THE DYNAMICS OF THE AIRPLANE
It is evident what we mean by the terms leading edge,
trailing edge, and nose.
8. Characteristic Curves for a Given Wing. By means
of experiments in a wind tunnel we investigate R, L, D as func-
tions of V and 6. We find that we can write, as for a flat plane,
= KAV 2 ,
where
= K y AV 2 ,
= K V 2 +K X 2 .
= K X AV 2 ,
The quantities K, K v , and K x are again constants for a given
angle and geometrically similar aerofoils. The 'values of the
coefficients K v and K X) and the ratio L/D = K V /K X for a certain
aerofoil * are given in the following table. The table also gives
the position of the center of pressure, which is considered in 12.
Angle of
Atack.
KV
K x
LID
Distance of
C. P. from
Leading Edge
-4
. 000399
.0001515
2.64
2
+ .000156
.0000905
1.72
I
.000432
.0000700
6-15
.620
.000721
. 0000653
11.00
530
I
.000936
.0000670
14.00
463
2
.001146
. 0000688
16.60
415
4
.001510
.0000860
17-50
340
6
.001878
.0001158
16. 20
.316
8
.002230
.0001558
14-30
303
10
.002580
.0002055
12.60
.290
12
.002910
.0002595
ii. 20
.283
14
.003165
. 0003040
10.40
.274
16
.003165
.0003710
8.50
.276
18
. 003080
.0005520
S-6o
.310
20
.002882
.0008500
3-40
.360
In order to use these values to determine the lift and drag,
the velocity V must be given in miles per hour, the area A
of the wing in square feet. The values of L and D that are
then given by the formula will be in pounds.
* This wing is U. S. A. No. i in the Third Annual Report of the (American)
Advisory Committee on Aeronautics. The shape of the wing is considered in 10.
THE PLANE AND CAMBERED SURFACE
9
The values of K v , K x and the ratio L/D are also plotted
in the following curves. It is to be noted that the vertical
scale is not the same for the three different quantities.
L
X
x
V
\
7
A
\
We note the following facts:
i. There is lift at an angle 2, i.e., sustentation exists ] or a
negative angle of attack.
10
THE DYNAMICS OF THE AIRPLA1
2. The lift increases almost as a linear f 'unction of the angle
and attains a maximum at about 15, then decreases rapidly.
j. The drag remains sensibly the same over small angles and
increases very abruptly in the vicinity 0/15.
4. The ratio L/D increases practically as a linear function for
small angles, and attains a maximum in the vicinity 0/15.
We shall merely note here the importance of the. ratio L/D.
For horizontal flight the lift must equal the weight of the
machine. Consequently the greater the quantity L/D the
Ky
.0030
.0020
.0010
C
x
.-
14'
-i!>
/
'12
/
4
4
/
/6>
/
/
r
r
r
V
*'
\
2
.0002 .0004 K,
FIG. 7.
less resistance there is to be overcome, and consequently the
less power is necessary for a given speed of flight.
It is from a study of the characteristic curves for a given
aerofoil that one decides upon its efficiency or suitability for
a given type of machine. It is not our purpose to go into
this question, and we shall merely remark that the type of wing
to be selected depends upon whether the machine is designed
for great speed, for rapid climbing, or for carrying heavy loads.
9. Polar Diagram. Instead of plotting the lift and drag
coefficients with the angle of attack as argument, we can plot
THE PLANE AND CAMBERED SURFACE 11
the lift against the drag. It gives, however, better results if
if we take different scales for K y and K x . We obtain in this
way the polar diagram, which has been extensively used by
M. Eiffel. In what follows we shall see its suitability for many
purposes.
10. The section of any wing depends upon the values of
BC, DC, EC, measured in terms of the chord AB, upon the
shape of the nose, and the general shape of the trailing edge.
Questions of strength and facility of construction are intimately
connected with those of thickness. By varying the different
elements one at a time, we are able to arrive at conclusions
as to the best value of any element, and determine the best
section for a given purpose. As an average we can state that
CB equal 3/8, and CD lies between 0.05 and 0.08. The effect
of the under camber is not so well known.
In Fig. 8 there is given the shape of the wing for which
the curves are given in 8.
It is also necessary to study the effect of the shape of the
ends of the wing. At the ends leakage occurs, and the air
flow is accordingly greatly modified. A wing with trailing
edge slightly longer than leading edge is found to be most
efficient.
For structural reasons the aspect ratio does not usually
exceed 8, as the advantage secured from increased lift is then
overbalanced by the increased weight of construction.
11. Pressure over the Wing. It is not only possible to
study the total resultant pressure on a wing, but also by intro-
ducing tubes through small holes in the surface, and con-
necting them to a manometer, it is possible to determine the
pressure at different points. The results for a central section
are shown in Fig. 9. It is found that the pressures on top
of the wing are below atmospheric. Consequently there is
12 THE DYNAMICS OF THE AIRPLANE
suction on top of the wing. The pressures underneath the
wing are greater than atmospheric, so there is an active upward
pressure. The two combine to make the total upward pressure.
In the figure, the suction on the top surface is represented by
lines drawn outward from it, and the pressure on the lower
surface, similarly by lines drawn outward. It is seen that
the suction is the greater of the two forces. In fact, it con-
tributes about three-fourths of the total sustaining force.
This also shows why there can remain sustentation for negative
incidence.
FIG. 9.
We find also in this way that we must not make the chord
too great, or towards the trailing edge we have pressures above,
and suction below, which would lessen the sustentation.
In the same way the pressures along any section can be
studied, so that the pressures all over the surface can be
mapped, and we obtain a very clear visualization of the air
reaction.
It should be noted here that the difference of pressures
from atmospheric pressure are very slight. But by having
large wing surfaces, and obtaining sufficient velocity we can
make the total lift equal to or greater than the weight of the
machine.
12. Center of Pressure. The resultant pressure is a vector,
and so is completely fixed in magnitude and direction. We
know all there is to know about it if we know its magnitude
and its moment about some point, for instance the leading
edge. It is, however, customary to speak of the center of
THE PLANE AND CAMBERED SURFACE 13
pressure, which we define as the point where the vector repre-
senting the pressure intersects the chord.
It is important for us to know how the center of pressure
behaves as the angle of attack changes. In general, it is found
for cambered surfaces and the angles employed in aviation
that the following is true:
The center of pressure recedes from the edge of attack as the
angle of attack diminishes. For larger and increasing angles
it recedes. This property should be contrasted with that given
above for plane surfaces.
Fig. 10 shows the motion of the center of pressure for the
wing considered in 8.
FIG. 10.
13. Relation between K y and K x . It is impossible to
obtain a simple relation between the coefficients K y and K x ,
but an approximate one will be obtained, and use will be
made of it later.
In the note to 4 there is given the formula of Colonel
Duchemin for the pressure on a flat plane as a function of the
angle of attack. It is obvious that for small angles it varies
approximately as the sine of the angle. Let us assume such a
relation for a cambered wing. As there is still lift for a negative
incidence we would expect to measure angles from the position
of the wing that gives no lift. We shall not do this, however,
and the results obtained will not be valid for the small angles
of attack.
We assume a relation
where K is the coefficient of pressure, and K' a constant.
14 THE DYNAMICS OF THE AIRPLANE
Assuming that the pressure is normal to the chord we would
then have
K y = K' sin e cos 6, K x = K' sin 2 0.
Hence
K v 2 K'cos 2 6'
If 6 is small this is approximately constant.
In order to see something about the accuracy of our result
we shall actually calculate the value of Kx/Kj 2 for the values
given in 8. The results are as follows:
Angle i 2 4 6 8 10 12 14
K*/Kf 76.40 52.3 47.4 33.6 31.3 30.8 30.6 30.3
While the value is not constant, it is seen that between 6 and 14
it is practically so, and approximate results can be obtained by
assuming that it is constant.
14. The Biplane. In order to secure greater lifting forces,
and at the same time deal easily with the problem of con-
struction, it is customary to
use two or more surfaces, one
above another. We shall limit
ourselves to the biplane.
By the gap is meant the
distance between the two
planes, measured in terms
of the chord, i.e., the ratio
BC/AB. By the stagger is
meant the distance CD, also
FIG. ii. , . r . _
measured in terms of AB.
The stagger is positive if the upper plane is in advance of the
lower, negative if in the rear.
The question of the relative efficiency of the upper and lower
wings enters at once. The gap is limited by constructional
reasons, and usually varies between i and i.i. It is found
then that the efficiency of the lower wing is considerably lessened
by the presence of the upper wing. The reason becomes
apparent when we recall what was said in u. The region of
THE PLANE AND CAMBERED SURFACE 15
depression above the lower wing is considerably modified by
the upper aerofoil. And as this depression is what gives the
preponderant part of the lift we would expect the lift to be
lessened. On the other hand the suction above the upper
wing is unimpaired, while the less important pressure below is
somewhat modified. The lower wing then has less efficiency
than the upper one, and on this account its aspect ratio is
sometimes made smaller than that of the upper wing.
The center of pressure for a biplane is purely a matter of
definition. We shall understand it to mean the point of inter-
section of the vector representing the pressure with a line parallel
to the chords and midway between them.
15. Body Resistance. In the forward movement of an
airplane, all parts of the machine contribute to the resistance
that must be overcome, while the aerofoils alone contribute to
the sustentation.* It is necessary to make the resistance of
all parts of the machine as small as possible. This is done by
giving proper shapes to fuselage, struts, wires, landing gear,
etc. The resistance that arises from the parts other than the
wings can be conceived of as arising from the motion of a
square plane normal to the direction of motion. We call this
the equivalent detrimental surface. Let its area be s; the
resistance due to the body can then be written
f=ksV 2 .
For the complete machine we therefore have
L = K y A V 2 , D = K X A V 2 +ksV 2 .
16. Experiments on Complete Model. We have thus far
discussed experiments made on the models of the wings.
Models of complete machines, with the exception of propeller,
wires, etc., whose contribution to the complete resistance is
small, are also subjected to exhaustive study in the wind tunnel.
Data necessary for the discussion of stability are obtained in
* The body doubtless adds something to the sustentation, but it is not an
amount of which we can take account, and is negligible when compared with that
furnished by the wings.
16
THE DYNAMICS OF THE AIRPLANE
THE PLANE AND CAMBERED SURFACE 17
this way. A diagram of a model of this sort is shown in Fig. 12.*
Lines showing the direction of the air reaction for angles of
attack varying from 1 to 8 are shown.
A question of primary importance at once arises. To
what extent, and under what conditions are the results obtained
from experiments on models applicable to full-scale machines?
The study of this question depends upon what is called dynamical
similarity. We shall not touch upon it here. In the appendix
reference will be given to treatments of this important question.
* This is taken from Hunsaker's paper, "Dynamical Stability of Aeroplanes,"
Smithsonian Miscellaneous Collections, Vol. 62. No. 5, 1916. It is a model of a
biplane tractor designed by Captain V. E. Clark, U. S. A., and is representative
of modern design.
CHAPTER II
STRAIGHT HORIZONTAL FLIGHT
17. LET us consider the motion of an airplane. Suppose
the motor is running. The machine is at any instant acted
upon by three forces, its weight, acting downwards through
the center of gravity, the traction of the propeller, and the
resistance of the air. As the machine is a rigid body we can
in general combine these three forces into a single force and a
couple. The force determines the instantaneous motion of the
center of gravity, and the couple determines the rotation which
the machine is momentarily undergoing. When regarded in
this general way the problem is seen to be very complex, on
account of the nature of the resistance of the air, for this
resistance depends upon the angle at which the wings are any
instant attacking the air, and on the velocity. Thus both the
motion of the center of gravity and the rotational motion
affect it.
In order to fix the ideas, suppose the airplane moves con-
stantly in a vertical plane, coincident with the plane of sym-
metry of the machine; then the center of gravity traces out
a certain plane trajectory. Suppose further that the angle
of attack remains constant; the angle between the tangent
to the trajectory and the chord of the wings is then constant.
The relative .wind at any instant is in the direction of the
tangent. Furthermore, it is evident that what we have called
the lift in the preceding chapter will now be in the direction
of the normal to the trajectory, and the drag will be in the
direction of the tangent. The motion of the machine will then
be determined by a force equal to the weight downward through
18
STRAIGHT HORIZONTAL FLIGHT 19
the center of gravity, the traction of the propeller along the
tangent to the path, the lift
2
K V A
9
perpendicular to the tangent, and the combined drag of the wings
and body
along the tangent to the path.
18. Horizontal Flight. We consider first the simplest
possible case. Let the machine be moving horizontally in a
straight line, with constant angle of attack. What are the
conditions that must be fulfilled F
to make this possible, and what
properties about the motion can be
discovered?
The lift L is now a vertical force,
and the drag a horizontal one. In
the figure, F represents the resultant
air pressure, applied at P, W repre-
sents the weight applied at the
center of gravity, and T the traction
of the propeller applied at a point
B, any point in the axis of the
FIG. 13.
propeller.
We are supposing that the machine is moving horizontally.
Hence W = L. Furthermore, we must have T = D, for if, for
instance, T>D, the velocity would start to increase. This
would increase L, and the machine would start to rise. Like-
wise, if T,
where has the meaning given to it in 23. At the optimum
angle we have ^ = and from the shape of the polar curve
we see that the point M 2 satisfying the last condition is to
28 THE DYNAMICS OF THP; AIRPLANE
the right of Mi, the point corresponding to the optimum angle.
The corresponding angle of attack, which we shall denote by a 2 ,
is called the economical angle. Flight at that angle requires
a minimum rate of consumption of fuel, assuming that the
efficiency of the propeller is practically constant over the
region considered.
It is not difficult to show that the equation last written
will also be true at the points M2 and Mi if the polar is con-
structed with different scales along K v and K x .
We can summarize the results of this section as follows:
The economical angle is greater than the optimum angle.
The machine flies faster at the optimum angle than at the
economical angle.
29. Although the consumption of fuel is at a less rate at
the economical angle, the machine flies more slowly, so con-
sumption takes place over a longer period, in case a given
distance is to be traversed. In fact the work per unit distance
traversed is exactly equal to the traction, and is therefore a
minimum when the traction is least. Furthermore, high speed
is usually a desirable feature, so flight at the optimum angle
is preferable to flight at the economical angle.
We can obtain some interesting results by comparing the
increased speed with the increased power necessary when
flying at an angle other than the economical one.
Let V2j PI represent, respectively, the velocity and power
for the economical angle, point M 2 on the polar, Fig. 17, V
and P the corresponding quantities for any other angle, repre-
sented by M on the polar, K y , 2 and K v , the corresponding lift
coefficients. Then
V 2 v K v ' P 2 #, tan0 f '
where < 2 and < are the angles in the polar diagram referring to
the conditions of flight. We can then write
V P (tan 02
V 2 P 2 tan
STRAIGHT HORIZONTAL FLIGHT 29
Let Mz be the other point where OMi intersects the polar, and
/ the corresponding angle of attack. Then if M lies between
Mz and Mz the quantity tan < 2 /tan is greater than unity.
Consequently, for angles of attack between a, and / the speed
increases at a greater rate than the power necessary. It is not
until the angle of attack falls below / that the increased
speed is obtained by a disproportionate increase in necessary
power, and therefore fuel.* We see also that the farther M dz\AK yP (z)) ds~ AK y dz\p(z))'
ds\af(s)
The traction can then be written
\m sin B d ( i \ , b~\
T=-Wcos6\ - -y-^ + tan0 .
L 2AK V dz\p(z)/ a\
Suppose that the angle of descent is given by the relation
tan 8 = -.
a
Before making the simplification that results from this assump-
tion we note that W/K y A is the square of the horizontal
velocity at the ground for the given angle of attack. Denote
it by F 2 . Then
V 2 m sin 20 d I i
4 dz
If the air were of constant density, we see that it wuuld
be possible for a machine to glide in a straight line without
any tractive force being supplied. For a given angle of attack
34 THE DYNAMICS OF THE AIRPLANE
there is a single angle of glide for which this is possible, namely,
the angle given above. It is seen to be the angle in the
polar diagram. In this case the component of the weight
along the path exactly balances the drag, and, if the machine
is started with the proper velocity along this course, it will
maintain the course without change of velocity.
With the air varying in density the situation is quite other-
wise. We have now for the square of the velocity
W cos S W cos 9
Sll
a/(s) AK v p(z)'
which decreases with the altitude. If the angle of descent is
that chosen above, the component of the weight along the
path balances the drag, and hence an outside force must be
used to cause the retardation. The necessary retarding force
is of course small, on account of the slow change of the density
of the air.
34. We have found the value of the traction with the assump-
tion of rectilinear descent. We must reverse the question and
assure ourselves that by starting from proper initial conditions
rectilinear descent will result if the proper traction is con-
stantly furnished.
We start with the equations
mu = W cos 0af(s)u,
as
m dli rr> , TT7- 7 ff \
= r+PFsm d bf(s)u,
2 ds
and suppose that the initial conditions, those for 5 = 0, satisfy
the relations
W cos6-af(s)u = o,
IF sin 0-bf(s)u = o,
that is, initially,
b W cose
tan 6 = -, u = 77-T
a af(s)
Put
_Wcos6
af( S ) '
DESCENT AND ASCENT 35
and suppose that the traction is given constantly by the expres-
sion
f j, = md^
2 ds'
The equations of motion can then be written,
dd T , 7
mu = W cos
ds
[" u\
i ,
v]
m d f \ TIT b ii\
(u-v) = W cos 6 tan 6
2 ds a flj
a v
Making use of the definition of z>, and putting
w = u v,
the equations take the form
dd ,, s
mu = awf(s) ,
(ts
2 as
The quantity u still occurs in the first equation, but will cause
no inconvenience. The initial conditions are = tan~ 1 -, w = o.
a
while u initially has a definite positive value.
Consider an interval o = s = s' so small that within it w
and 6 have no extrema, and therefore their derivatives are
not zero except for s = o. Therefore throughout the interval
w and 6 are mono tonic.
Suppose w were increasing, so that ->o. Then since wf(s)
Q/S
is increasing, and both terms on the right-hand side of the second
equation are initially zero, we see from that equation that d
must increase. Hence >o. But this evidently contradicts
ds
the first equation.
36 THE DYNAMICS OF THE AIRPLANE
Suppose next that in the interval considered w is decreasing,
and therefore negative, since it is initially zero. The second
term on the right of the second equation will be >o in the
interval, so that we must have tan 0<-, for -7- must be O. There is
ds
therefore again a contradiction.
It follows that throughout the interval we must have w = o.
Therefore
=
af(s)
It also follows that
6 = constant = tan" 1 -.
We see that the initial conditions that we have assumed
are maintained throughout the interval o = s = s', and therefore
by a process of continuation will be maintained throughout
the motion. Therefore rectilinear motion will assuredly result
from the initial conditions and the value of the traction that
we have assumed.
35. We shall next consider the path of descent that is
followed when the machine is descending with the propeller
not running. The equations of motion are
de
mu = W cos 8af(s)u.
m = 2W sin 62bf(s)u.
We have already demonstrated that, if the density of the air
remained constant, it would be possible to glide down a straight
line, provided the machine started in the correct direction with
the proper velocity. In the present problem we shall assume
these same initial conditions. That is, for s = o, we assume the
right-hand sides of the two equations to have the value zero.
We recall further that/(Y) is an increasing function.
DESCENT AND ASCENT 37
We consider as before an interval o = s = s' within which
u and 6 are mono tonic. Suppose u increases; then T~>O,
QS
and the second equation shows that 6 must increase, that is,
>o. But from the first equation, since 6 increases cos 6
ds
decreases, and we see that the right-hand side, being zero initially,
becomes negative. Hence -ro. In order
ds
that 9 may be increasing the second equation shows we must
assume f(s)u to be increasing. The first equation then has its
right-hand side negative, since cos decreases, and f(s)u
increases. This gives a contradiction, since with 9 increasing
we have - >o. Suppose, now, that 9 decreases; then f(s)u
ds
may be either increasing or decreasing so far as the second
equation is concerned. In this instance -T- let us use the ordinary polar angle,
namely, the angle that the radius vector makes with the -K^-axis,
and which we denote by ft. Since sin = cos ft, we see that
v will be a minimum when r/cos 2 ft is a maximum. If we knew
the equation of the polar curve, the point for minimum v
would be obtained by equating the derivative of f/cos 2 ft to
zero.
We next consider the angle that gives minimum power.
In 28 we have shown that we can write
where A is a constant, and x and y are the Cartesian coordinates
of a point on the polar, with Q for origin. When transformed
to polar coordinates this becomes
\/ * 3 *
To find the angle of attack for minimum power, it is then
sufficient to find the point on the polar for which
r sin 3 ft
cos 2 ft '
is a maximum. The derivative of this with regard to ft can be
written
in 2 /?
Return to the point on the polar that corresponds to the
angle as for minimum vertical velocity. For that point we
have
*( r \ =Q
DESCENT AND ASCENT
41
Furthermore, this value of (3 gave a maximum to the function
r/cos 2 0. Consequently,
d r
according as is greater or less than the value fa corresponding
to 0:3. Now we know that az is not much different from 2,
which is known to be greater than ai, the optimum angle.
Assuming then that a 3 >i we see that on the part of the curve
FIG. 19.
corresponding to a>a\ the polar angle ft is a decreasing function
of a. Suppose then thata>o;3. It follows that @<(3z, and con-
sequently,
dfr\'
I 1 >o.
d/3\cos 2 j(3/
If, however, ai03, and consequently
d( r \
- Ps and
consequently
FIG. 20.
and M 2 could lie some place between MS and M 3 '. It could
not lie to the left of Ms or the right of MS, as in either case
j8
The results that we have derived also show that the traction
necessary to mount is the same that it would be if the machine
were subject to two forces, its weight and a horizontal force T.
The traction will be a maximum when the flight path is
along the diagonal of the rectangle. In this case = 90 0,
and
T
FIG. 22.
T =
W
sin cos
If we take .i5 = tan0 as a medium value of the fineness,
we find for the maximum traction i.oiXW, that is, the maxi-
mum traction to be experienced in ascending is i per cent
greater than the weight of the machine. As long as the angle
of ascent is less than 90 20 the traction is less than the
weight of the machine. This angle can be as high as 76,
when the fineness is close to .12. As the angle 6 continues
to increase from this point the traction exceeds the weight,
reaching the maximum given above at a value as high as 83
for the angle of ascent. It then decreases as the angle in-
creases to 90, on account of the rapidly decreasing speed,
46
THE DYNAMICS OF THE AIRPLANE
when it again equals the weight, and we have the case of motion^
less sustentation.
It is obvious that for ordinary angles of ascent the traction
will not exceed the weight.
43. The Power Necessary. The power necessary to ascend
On the diagonal of the rectangle used in the last section
as diameter construct a circle,
omitting that part of the circle
that lies below the horizontal and
the part to the left of the vertical.
On the vertical lay off a distance
V and to the left of the verti-
cal construct the velocity curve
FVcos 0, as shown in Fig. 23.
For a given angle of ascent the
power is then given by the area
of the rectangle OABC.
Let the diameter of the circle be D, and 7 the angle shown.
Then
P = D cos 7-FVcos 0.
Hence,
_D-V cos 7
dO 2 Vcos
= -D-V sin yVcos -?-
sin 6.
Evidently dj/dd=i. so that the condition for a maximum
(the minimum of P is o, for B = 90, for then V = o) of P is
tan 7 = J tan 0.
For this relation to be satisfied it is evident that 0<9O .
Therefore, the angle for ascending at which the power is a
maximum is less than the angle for which the traction is greatest.
If 61 is an angle for which the power necessary to ascend
is greater than the power necessary for horizontal flight,
there is a second angle 02 for which the required power is the
same as that for B\. The velocity for 2 being less than for 0i,
the traction must be greater. (We note, however, that we
DESCENT AND ASCENT 47
cannot say that simply because 02>0i the traction corre-
sponding to 62 must be greater than that corresponding to 61,
because the traction has been shown to have a maximum.)
44. The Vertical Velocity. The rapidity with which the
machine is momentarily moving vertically is
z;=F-sin 6.
If the equation of traction is multiplied by V we can write
W-v = P-(K x A+ks)V*,
or in terms of V,
W-v = P- (KA +ks) F 3 cos 3 '* 6,
which can again be simplified into
where P is the power necessary for horizontal flight.
If the angle of ascent is small, we can write
P P P P
^ = 375 TTr miles per hour = 5 50 - feet per second,
Vv W
where P and P are measured in horse-power.
Now P is the useful power that the motor is developing,
so that, providing the angle of mounting is small, the velocity
of ascent is equal to the excess of power over that required for
horizontal flight, divided by the weight. Further, we see
that for a constant available power, the velocity of ascent is
greatest when the angle of attack is a t , that is, the angle for
minimum power for horizontal flight. Remembering the
results that were obtained for minimum vertical velocity in
gliding, we see that an angle of attack that gives a small vertical
velocity in gliding, will give a large one in ascending, provided
we have a constant available power.
The ideas that have just been adduced must be modified
when we come to consider the propelling plant. Changing the
angle of attack changes the velocity and traction. This alters
the number of revolutions per minute that the propeller must
make in order to furnish the traction at that velocity. This in
48 THE DYNAMICS OF THE AIRPLANE
turn alters not only the efficiency of the propeller, but also
the power which the motor is developing. Thus in practice
we do not have a constant available power. Consequently
the best angle for mounting cannot be expected to be exactly
the economical angle.
Another approximation is useful. We have
v v
? fc^*V
We are assuming that practically we can take cos0 = i, and
can therefore take 8 = sin 6. In degree measure we can there-
fore write
P-~P P-~P
approximately, provided the result is a small angle. In this
formula the excess useful power is expressed in horse-power,
the weight of the machine in pounds, and the velocity in hori-
zontal flight in miles per hour.
45. We return to the general considerations of 39. Suppose
the pilot upon leaving the ground gives full admission of
gasoline, developing more power than is necessary to fly hori-
zontally near the ground at that angle of attack. The machine
starts to rise. The values for the velocity, traction, and power
which have been obtained in our discussion are those that
exist at any moment. They are known if the direction of the
tangent to the path is known. The general problem that arises
is of great difficulty. The power of the motor for a specified
number of revolutions per minute decreases as the machine
gains altitude, because the diminishing atmospheric pressure
decreases the mass of gas mixture that is in the cylinder at
each explosion. Further, the changing density of the air
modifies the propeller's action, and the changing velocity
alters the number of revolutions per minute. If the angle of
attack is not changed, the curvature of the path decreases
with the altitude, and the machine ultimately flies horizontally.
This occurs when the power the motor is developing multiplied
by the efficiency of the propeller is exactly the power required
DESCENT AND ASCENT 49
for horizontal flight at that altitude. The height to which the
machine has risen is called the ceiling. It is dependent upon
the angle of attack. But the word can be used, without danger
of confusion, to denote the height to which the machine can
rise for a given angle of attack, or the maximum of these heights,
that is, the greatest height at which the machine can possibly fly.
As the machine is fitted with various instruments for
measuring height, velocity, time, etc., a record of the flight
can be made, and from such data experimental laws can be
adduced. Though some of these laws may appear only approx-
imate, they will nevertheless allow us to obtain, by equations
derivable from them, other results of sufficient accuracy to be
both of interest and value. And it is only in this way that cer-
tain problems can be attacked.
46. Experimental Law of Vertical Velocity. Experience
shows that the vertical velocity decreases sensibly as a linear
function of the altitude. Suppose the machine is flying at the
angle of attack that allows it to reach its ceiling. Let h be
the height of the ceiling, and VQ the vertical velocity at the
ground. Then
where z is the altitude. It is convenient to measure the
velocities in feet per second and // and z in feet.
Experience also shows that the initial velocity ^o can be
calculated very approximately from a knowledge of the power
of the motor, the weight of the machine, and the height of the
ceiling. Let HQ be the power of the motor at the earth, using
full admission of gas. Then the quantity
W
W== JT
#o
is the weight per unit horse-power, a quantity, like the loading,
of great importance. The quantity VQ is then given approx-
imately by the relation
h
'
50 THE DYNAMICS OF THE AIRPLANE
For example, a machine weighing 7.59 pounds per horse-
power, and having a ceiling at 24,500 feet, will 'have an initial
vertical velocity of 34 ft. /sec.
We can then write
h-z
v = -- .
95*-
47. Time of Ascent. We have
dz
5
Integrating this we find
*()-*. log
.-
B)
In terms of common logarithms this becomes
h . i
= 2. 303 --log
VQ
the time t(z) being expressed in seconds.
48. Determination of the Ceiling. The result which has
been derived for the time of ascent can be used in a very simple
way to determine the height of the ceiling.
We have
H)-
Let /(zi) and t(z%) be the times required to reach altitudes
0i and 22, respectively, zi =
g g AK V AK y '
This is consequently an inferior limit to the radius of the
turn that a machine can make. It depends upon the angle
of attack. It is smallest for the angle of attack that gives the
maximum value to K y . It is obvious that as the radius of the
turn approaches its inferior limiting value the angle 6 approaches
90 and the velocity V approaches infinity.
In practice, however, a machine has an inferior limit to the
radius of a turn it can execute that is much greater than the
theoretical one which has been derived. For long before the
theoretical limit has been reached, the power of the engine will
have become insufficient to furnish the necessary velocity.
52. The Traction and Power. The traction_and power are
easily expressible in terms of their values T and P for horizontal
flight at the same angle of attack.
We have
f ~V 2 cos 0*
Hence
i
In the same way
P TV i
p TV
so that
56 THE DYNAMICS OF THE AIRPLANE
Expanding the quantity on the right we have
o v 4 2i F
-4-2+-
4 r 2 g 2 32
Hence
Suppose the radius of the turn is large. Then at the same
angle of attack the increased power necessary varies approxi-
mately as the inverse of the square of the radius.
53. The results that have been derived have merely shown
how the correct turning force can be obtained by giving the
proper inclination to the plane of symmetry. It is necessary
to see how the machine can be gotten from its horizontal straight
line path into the required banked position, and how the tend-
encies to depart from this position can be overcome.
The controls that are used for the purpose are the rudder
and the ailerons.* Suppose the rudder is turned towards the
right.f A moment is produced turning the machine in that
direction, and at the same time a small force tending to make
the center of gravity move slightly to the left. The ultimate
effect of the air pressure on the rudder and the keel surface
is to start the machine turning towards the right. This causes
the left part of the machine to move faster than the right.
Consequently, the air pressure on the left is greater than on
the right, and the machine starts to bank towards the right,
which is the correct direction to produce a turn to the right.
This banking is not equal in general to that required for a
correct turn. The pilot increases the degree of bank by the
ailerons. The ailerons on the left are depressed and those
on the right are raised, thus producing a greater lift on the left
wing. The machine having been banked, the lift on the wing,
which before the turn was vertical and just equal to the weight,
now departs from the vertical, so that the component that acts
in the direction opposite to gravity is no longer sufficient to
* For a description of the ailerons see 98.
t See 99, 100 for a fuller explanation of the rudder and its action.
CIRCULAR FLIGHT 57
sustain the machine, and the machine will start to Jive, unless
its velocity is increased. For this purpose the pilot may either
increase the admission of gas, or alter his angle of attack, so
as to obtain an excess of power. If it should happen that his
machine is " tangent/' that is, an excess of power cannot be
secured in either of these ways, he cannot turn, without de-
scending to a lower altitude, and thus liberating some excess of
power.
As the outside of the machine moves faster, the drag there
will be greater than on the side towards the center of the turn.
This has a tendency to make the machine turn towards the
left, and it must be overcome by the controls. The turning
moment on the machine necessary to keep the axis along
the path is secured from the rudder.
We have seen that in order to turn correctly it is necessary
to produce a turning moment and a centripetal force. In
general, we can say that the function of the rudder is to produce
the necessary turning of the axis of the machine, and the
function of the ailerons to give the banking that is required
to produce the necessary centripetal force.
It is also by means of the ailerons that a tendency to slip
is overcome. If the machine tends to slip towards the outside
of the curve the ailerons on that side are depressed and those
on the other side raised. This banks the machine further
towards the inside, throwing the resultant air force further from
the vertical, and thus increases the force towards the center.
In case the machine starts to slip towards the inside of the
curve the ailerons are manipulated in the contrary way.
When it is desired to straighten a machine out after a change
of direction has been accomplished it is necessary to make opera-
tions the reverse of those described for making the turn.
When making a turn the pilot must guard against heading
upward, because the loss of speed may be too great to insure
stability, unless a large excess of power is available.
It is evident that in a banked position the function of the
elevator and the rudder are not distinct, but have closely
related effects, depending upon the degree of inclination.
58 THE DYNAMICS OF THE AIRPLANE
In some early machines, turning was effected without
banking the machine. The necessary centripetal force on the
center of gravity was secured by the air pressure on vertical
partitions. When the rudder was in a neutral position, there
was no pressure on these surfaces, but when the rudder was
turned, a force arose that produced the turn. A good deal
of slipping arose in such a 'turn, and therefore the turn could
not be considered as correct. Furthermore, the drag on
the vertical surfaces reduced the speed of the machine.
2. CIRCULAR DESCENT
64. We shall next consider the question of helical descent,
without the motor running. The traction necessary to over-
come the drag of the machine will be furnished by the weight
of the machine, and the necessary centripetal acceleration will
be derived from the air reaction, the machine being again in a
banked position. We shall assume that the air density can be
considered constant in the distance through which we are con-
sidering the motion.
55. Let AGB represent the path of the machine, G being
the position of the center of gravity at any instant. Let
r represent the radius of the cylinder on which we are supposing
the helical path is traced. Let CGD be the horizontal circle
through G. The weight of the machine is represented by
GW = W, drawn vertical. The centrifugal force is repre-
sented by GH = H, outward along the radius. The resultant of
these two forces is represented by GQ = Q, in the plane through
the axis of the cylinder and point G, and making an angle
with the vertical. The air reaction must then be represented
by GR, opposite in direction and equal numerically to GH.
The velocity of the machine is represented by GV=V, the
horizontal component by GVo = VQ. Let further \l/ be the angle
between GV and GV .
In order that the turn may be properly executed it is neces-
sary that the axis of the machine coincide instantaneosuly
with the tangent to the path. Therefore the plane of sym-
CIRCULAR FLIGHT
59
me try is determined by GR and GV. The lift component of
the air reaction lies in this plane, being perpendicular to GF,
and the drag is along GV. The resultant Q of W and H also
lies in the plane of symmetry. We can then consider that the
force Q, the lift L, and the drag D are in equilibrium. In the
FIG. 25.
plane of symmetry draw a line perpendicular to GQ. Let the
angle between it and GV be ^ '. Since the lift L is perpen-
dicular to GV, we have, upon resolving forces perpendicular
to and along GV } the following equations:
60
THE DYNAMICS OF THE AIRPLANE
as the equations of sustentation and traction, respectively.
The angle 6 is evidently given by the relation
tan , = ^ 2 = zo? , '.;;
Wr gr
where m is the mass of the machine.*
56. The angles 0, \J/ and ^' are evidently not independent,
and the relation between them can be easily found by spherical
trigonometry. Let GZ represent the vertical through G, GX
904-0
FIG. 26.
FIG. 27.
the tangent to the circular section CGD, and GY the radius
continued outward. Draw GZ' in the plane of FZ, making an
angle with GZ. Likewise draw GX' in the plane of XZ,
making an angle \f/ with GX. Then evidently the plane of GZ f
and GX' represents the plane of symmetry of the machine.
About G describe a sphere of unit radius. We thus have a
right spherical triangle which we can represent separately from
the axes, as shown in the Fig. 27. We have then
cos C = cos 0-cos (90+^).
If we recall the definition of ^', we see that it is the angle
between GX' and a line in the plane of GZ' and GX' and per-
pendicular to GZ'. It then follows that
Devillers, "La Dynamique de PAvion," p. 135.
CIRCULAR FLIGHT
61
The preceding relation then becomes
sin \l/' = cos 6 sin ^,
from which we note in passing that
We can also find the inclination of the plane of symmetry.
This inclination is the angle between the plane of symmetry
and the vertical, that is, the angle / in the right spherical
triangle we have constructed. Therefore,
tan/ =
tan B
cos \l/'
This relation can be put in another form which will later
be useful. We have
cos I
cos \f/
cos
Therefore,
Hence,
Vcos 2 ^-ftan 2 B Vsec 2 6 -sin 2 ty
_ cos \f/ cos 6
Vi sin 2 \l/ cos 2 6
cos/ =
cos ^'
cos 6
cos
cos \f/ f
cos
cos /'
57. Other Form of Equations of Motion. We can put the
equations of motion in another form.
The path described is a skew curve,
whose osculating plane contains the
tangent and the radius of the cylin-
der. The principal normal is there-
fore along the radius; the binormal
is perpendicular to the tangent and
the principal normal.
We shall resolve the forces along
the tangent, principal normal and
binormal, represented, respectively, by GX, GY, and GZ.
FIG. 28.
62 THE DYNAMICS OF THE AIRPLANE
As the weight of the machine acts vertically downwards,
its components are respectively,
W sin ^, o, W cos ^.
Consider next the air reaction. It acts in the plane of
symmetry, which passes through GX and makes an angle 7
with the vertical plane XGZ. The lift L lies in this plane,
and is perpendicular to GX. Therefore its components are
o, L sin /. L cos /.
The drag is along OX, and its components are
-D, o, o.
Finally, we have to consider the centrifugal force, mVo 2 /r,
where VQ is the horizontal projection of the velocity. This
force is along the radius, so its components are
- , o.
For steady flight the sum of the forces along each axis
must be zero. Hence we have:
D = W sin \j/,
mV 2
Lsm/ = ,
L cos / = W cos \l/.
If we divide the second equation by the third, we can take,
as the equations of motion : *
D = Wsmt,
LcosI = W cos \l/,
tan/-- 5V-.
rg-cos ^
58. We next show that the equations of motion last derived
are identical with the first. In the latter we replace Q by
TF/cos 6, and insert L and D on the right-hand sides of the
equations of sustentation and traction, respectively. The
* Cowley and Evans, loc. cit, p. 222.
CIRCULAR FLIGHT 63
former equations then become, when written in the order of
those in the last paragraph :
COS0
W
If now we make use of the relations:
sin \l/' = cos 6 sin \f/,
cos \f/' _ cos ^
cos 6 cos I y
tan 6 = tan 7 cos ^,
which were derived in 56, we see that the equations of motion
last written become identical with those obtained in the last
paragraph.
59. From the equations of sustentation and traction we have
so that \l/' is equal to the angle of glide for rectilinear descent,
and is therefore uniquely determined by the angle of attack.
We shall again regard r as an independent variable, specify-
ing the conditions of flight, as in the case of circular flight, and
determine the other quantities that specify the motion, namely,
\j/ and B in terms of it and ^', which we have just seen is deter-
mined by the angle of attack.
When we substitute
F =Fcos f,
in the third equation of motion we obtain
V 2 cos 2 t
tan 6 = --- -.
64 THE DYNAMICS OF THE AIRPLANE
Making use of the equation of sustentation this becomes
rgK y A
But
cos 19'
so that we have finally
sin 6 = : - cos \j/ f cos 2 \f/,
m
r^-r COS
,/r sin 2 *'l
L 1 o5F*J'
upon making use of the relation between \f/, \J/' and S.
When we replace sin 2 \l/ f by i cos 2 ^', and cos 2 by i sin 2 9
the last relation reduces to
. o - w cos ^' . 9 . . . . w q . ,
sin 3 -- -- sin 2 0-sm ^H cos 3 ty =o,
a cubic equation for sin 0, in which all the coefficients are
known as soon as we know the radius of the helix and the
angle of attack.*
Put
m cos ^
The equation then reduces to
y? ax 2 x+a cos 2 ^ = 0,
where x is the unknown, and a and \j/ r are known. This equa-
tion can be written
(x a) (x 2 i) = a sin 2 ^',
or
/ _
1 "
We construct now the curve,
_sin 2 //
* The cubic equation for sin is given by Devillers, loc. cit., p. 138. His
discussion of the equation is quite different from that given here.
CIRCULAR FLIGHT
65
which consists of the three branches shown, asymptotic to the
lines #+i=o, x i =o. Draw finally the line
which passes through (0,1) and has a slope of i/a.
FIG. 29.
The roots of the cubic then correspond to the abscissae of
the points of intersection of the line and the curve. It is
obvious that all three of the roots are real. Let them be
denoted by xi, xz, and #3. It is seen that
- 00
Since # = sin 0, the only root which is applicable is X2. We
thus have sin 0, and consequently 0, uniquely determined.
66 THE DYNAMICS OF THE AIRPLANE
The diagram can be easily modified so as to allow the value
of to be directly obtained. Draw the curve x = sin 0, taking
for the negative ^-axis, as shown. By continuing the ordinate
through x = X2 until it intersects the curve # = sin d, the value
of B can be immediately obtained.
It is interesting to consider the limiting values for B when r
approaches zero and infinity, respectively . The corresponding
values of a are evidently oo and o. The slope of the line we
have used becomes oo for a = o, and evidently the corre-
sponding value of X2 is zero. Thus for r = oo we have 6 = o,
and the descent becomes rectilinear.
Let r = o, so that a =00. The root #2 is then determined
by
^sin 2 ^
IX 2 2
This gives
sin 2 6 = i -sin 2 f = cos 2 tf/,
and therefore
= 9 o-t//.
This is a maximum value for the angle 0.
After we have obtained the values of \l/ r and 6, we can
at once get ^, the angle of descent, from the relation
sin \f/'
sm ^ .
cose
In the limiting case r = o we have sin ^ = i, so that ^ = 90.
The machine in this case descends vertically, rotating about
its axis as it descends. The value of / is seen to be equal
to 90.
60. The Velocity. We have from the first two equations
of 55
Q
Hence,
CIRCULAR FLIGHT 67
Let V be the velocity for a rectilinear descent at the same angle
of attack. From 36 it follows that
V' 2
COS0'
and therefore V can be determined as soon as has been found.
We see that the velocity in a helical descent is greater than in
a rectilinear glide. In the limit for r = o, we have cos = sin \l/'.
Hence the limit of V is given by
V' 2
sin*'
where B is the fineness.
The distance that the machine will advance downwards for
each revolution about the axis of the cylinder, that is, the pitch
of the helix, is
h = 2irr - tan \f/.
In the limit as r approaches zero, this becomes indeterminate,
for \f/ then approaches 90. In order to investigate the limit
we shall express both r and \f/ in terms of 6 and \l/' '. We have
when this is done:
sin \l/' sin \l/'
tan \(/
Vcos 2 sin 2 \// f Vcos 2 fy' sin 2
From 59 we find,
_m cos \j/' cos 2 \}/' sin 2
K V A sin 0- cos 2 '
Hence,
, _ irm sin 2 ^' Vcos 2 \j/' sin 2
1^^ sin cos 2
In the limit when r = o we have sin = cos \j/'. Hence in the
limit h = o.
If we write,
h
- = 27r tan \f/,
THE DYNAMICS OF THE AIRPLANE
and note that when r approaches infinity the angle ^ approaches
^', we see that the graph of h as a function of r approaches the
line
h = r-2ir tan \/' = 2irB r
asymptotically.* But we know that t'. The resultant air
reaction R will be in a direc-
tion near to the normal to
It will have two components,
FIG. 32.
the chord of the section.
* For a discussion of various designs in propellers, see the report by W. F.
Durand in the Third Annual Report of the National Advisory Committee on
Aeronautics, 1917, Report No. 14, entitled "Tests on 48 Model Forms of Air
Propellers with Analysis and Discussion of Results and Presentation of the same
in Graphic Form."
THE PROPELLER 71
one in a direction opposite to the rotation, and the other along
the axis. The first component opposes the motor couple, and
the second produces a thrust along the axis. A similar analysis
holds for each element of the blade, and we' see that the resultant
effect of the whole propeller is the production of a couple that
must be overcome by the motor, and a thrust along the axis
of the rotation.
65. Geometrical Pitch. In the figure let = 0+0. The
value of 6 depends only on the inclination of the section, which
we shall call the setting of the section. A consideration of
the figure leads us to regard the chord of the section as a part
of a geometrical helix of radius r, that makes an angle 6 with
a section normal to the axis of the helix. From this viewpoint
one revolution should make the element advance a distance
H = 2irr - tan 6.
This is called the geometrical pitch of the element. It will
vary from element to element, unless the angle 6 for an element
is connected with the radius of the element by the relation
6 = tan- 1 ,
2irr
where H is some constant.
There are then two classes of propellers: those of constant
pitch, and those of variable pitch.
(i) Propellers of Constant Pitch. By choosing the angle 6
to satisfy the relation given above the pitch of the propeller
will be made constant. The angles at which various elements
of the blade are set are easily shown graphically. We con-
struct the distance H/2ir as an ordinate, giving the point P,
and on the axis of abscissae lay off distances equal to different
fractions of the propeller's radius. The corresponding points
are joined to P. The lines obtained evidently give the inclina-
tion of the chords of the various sections to a plane normal to
the axis of the propeller.
Suppose the propeller is of constant pitch, and that it
72
THE DYNAMICS OF THE AIRPLANE
advances due to the thrust it creates at the velocity V. We
have then for the angle 0, Fig. 32,
V
~ 1 --
V
- 1 -
2irnr
so that the angle of attack is
H
= 0- / 3 = tan~ 1 --
2irr
tan
, 2irr 2irnr nH V
- 1 ^tan- 1 -
H-V
(27rr) 2 n+HV
FIG. 33.
This depends upon r, so the angle of attack is different for
different elements of the blade, being the smallest at the tip,
and increasing towards the axis. This fact can be considered an
objection to a propeller of constant pitch, although the ultimate
decision must depend on experiment.
(2) Propellers of Variable Pitch. The angle can be chosen
so that the angle of attack of all elements will be the same,
by merely determining 6 by the relation
0+ tan
- 1
2irnr
where is constant. The manner of graphically obtaining the
inclination of the various sections is shown in Fig. 34. The
THE PROPELLER
73
ordinate is constructed of length V/2irn. Lines are drawn to
various points on the axis of abscissae as in the former case.
An angle equal to the chosen angle of attack is laid off from
each of these lines, and the new lines obtained represent the
chords of the various sections.
It is to be observed that if the angle of attack of all elements
of the blade is the same for a certain value of V and n, it need
not be the same for all elements for other values of V and n.
Let the propeller be designed for a velocity Fo, and number
of rotations no, and choose < as the angle of attack of all
elements; then the setting of an element is given by
V
2-mi
FIG. 34.
as a function of r. Now suppose this propeller advances at
rate V and makes n turns per minute. The angle is now
2irnr
and the angle of attack is
tan
It is seen that the necessary and sufficient condition that this
does not depend upon r is
nV noV = o,
that is,
V = Vo
n no
74 THE DYNAMICS OF THE AIRPLANE
The angle of attack will thus again be the same for all
elements of the blade, only if the advance per revolution is
the same as it was before. This quantity is of fundamental
importance in considerations regarding a propeller.
66. Thrust and Power. In order to calculate the thrust that
a propeller will produce, the power that is necessary to rotate it,
and the efficiency with which it is working, experiments are made
on model propellers, over a wide range of forward velocities, and
speeds of rotation. From the data thus obtained the action of
a full size, geometrically similar, propeller can be deduced.
We proceed to a discussion of the conditions under which we
can compare the action of similar propellers.
Consider geometrically similar propellers of diameters D
and DI, respectively. Let them be advancing at velocities
V and Vij and rotating n and n\ times a second, respectively.
In case the angles of attack of corresponding sections of the
two propellers are the same, we can compare the action of the
two; and it is quite apparent that we cannot expect to find
any simple relation between the action of the two propellers
unless this is the case. The condition for the equality of the
angles of attack of corresponding sections, always under the
hypothesis that the propellers act on air at rest, is easily
obtained. The settings of the corresponding sections are equal,
since the propellers are geometrically similar. It is then neces-
sary that the angle /3 be equal in the two sections. The con-
dition for this is obviously
V = Vi
nr n\r\
where r and r \ are the radii of the sections. This reduces to
nD
since r/r\D/D\ for similar sections.
It is assumed that the propellers are acting under conditions
that satisfy this relation.
In the two propellers take sections near those already
chosen, the new sections also being at distances proportional
THE PROPELLER 75
to the diameters. Consider the portions of the two blades
thus obtained as small aerofoils. As they are engaging the air
at the same angle of attack the air reactions will be assumed
to make the same angles with their chords, and thus with the
axes of the propellers. Let the air reactions be dF and dF\.
Then
dF ds-V' 2
where ds and dsi are the areas of the two sections of blade,
and V and V\ their total velocities (resultants of V and 2Trnr }
and Vi and 2-n-rini, respectively).
Now
in virtue of the similarity of the propellers, and
V 1 _ zirnr sec /3 nr nD
Vi 2^n\r\ sec |8 n\r\ n\D\
Hence
dF
Let dT and dT\ be the elements of thrust furnished by the
two sections. Since the reactions make equal angles with
the axes, we have
dT n 2 D*
By dividing the two propellers up into corresponding small
sections and adding the thrusts of the sections, we have
T
for the relation between the total thrusts.
This relation leads us to write
where a is some function of V/nD, applicable to geometrically
similar propellers. Such a relation must be confirmed by
experiment. This question will be considered presently.
76 THE DYNAMICS OF THE AIRPLANE
Let dR and dRi be the resistance to rotation of the two
sections. Then
dR
Let dP and dP\ be the work done on the two sections in a
second; then
dP = 2-wnrdR, dPi = i-Kn\r\dR\.
Hence,
dP
dPi
We are thus led to put for the power necessary to turn the
propeller
where /3 is a function of V/nD applicable to geometrically similar
propellers.
67. The results obtained can be put in another form. We
have
/^27^2\
D 2 .
Since a is a function of V/nD, the quantity a(n 2 D 2 /V 2 ) is
also a function of V/nD. Hence we are led to write,
T = 3V 2 D 2 (pounds),
where 3" is a function of V/nD.
In the same way we obtain
P = (P V 3 D 2 (foot-pound-seconds) ,
where (P is a function of V/nD.
68. To investigate the accuracy of the results that have
been obtained it is necessary to test several propellers that
are geometrically similar. Consider one of them. The thrust
T is measured for a large range of values of V and n. The
quantity T/V 2 D 2 is calculated and is plotted against the
argument V/nD. In this way a curve is obtained that repre-
sents the value of SI for the first propeller. The other pro-
pellers are then tested, and it is seen how nearly identical the
curves they give are with the first.
THE PROPELLER 77
In the same manner the formula obtained for the power is
subjected to verification.*
69. Efficiency. The efficiency of a propeller is the ratio
between the useful power it yields and the power that it absorbs
from the motor. Let it be represented by E. Then,
E= T ' V
550 -P*
the thrust T being measured in pounds, and the velocity V in
ft./sec. and P in horse-power. When the expression obtained
for T and P in terms of V and D are used this becomes
E being, as indicated, a function of V/nD, applicable to similar
propellers.
70. Effect of Altitude. The thrust that a propeller will
produce and the power necessary to turn it vary directly as
the density of the air, and will thus depend upon the altitude.
It is apparent, however, that the efficiency is independent of
the altitude.
The diagrams that will be given for a propeller's action
are for the surface of the earth. The thrust and power for a
given altitude will then be found by multiplying the thrust
and power at the surface of the earth for the same velocity
of advance and number of rotations per minute by the ratio
of the air density at the given altitude to the density at the
surface of the earth.
71. Graphs of the quantities 3", (P, E for a model propeller f
* Results of experiments of this sort on four geometrically similar propellers
of diameters 30, 36, 42, 48 inches, respectively, are given in Report No. 14, part I,
Figs, u, 12, of the Third Annual Report of the National Advisory Committee
for Aeronautics, 1917.
f This is propeller No. i, in the Report of the Advisory Committee on Aero-
nautics above referred to. The curves are not given in the form in which they
occur in the report. The density of the air at the surface of the earth is intro-
duced. The constant 100 that occurs in the report is omitted from the thrust
curve. The power curve is obtained from that for torque.
78
THE DYNAMICS OF THE AIRPLANE
are given in Fig. 35. The quantities 5" and (P are called the
thrust and power coefficients, respectively.
As an example of the use of the graphs, consider a full-scale
similar propeller with a diameter of 8 feet. What will be its
thrust and the horse-power required to turn it at an altitude
of 2000 feet, if it is advancing at a velocity of 72 miles an hour,
and making 1200 turns a minute?
.0009 .0018
FIG. 35.
V
We have 7 = 105 ft./sec., n = 2o. Hence = .66. From
nD
the diagram we find ^ = .00054, (P = .oooj. The ratio of the
density of the air at the given altitude to the density at the
surface of the earth is .9. We thus have
= 35o pounds,
.9X.ooo7Xio5 3 X8 2
= 85 horse-power.
550
THE PROPELLER
79
The efficiency can be immediately obtained from the graph,
arid is E = .75. It can also be computed directly from the values
for T and P.
72. A problem of fundamental importance is that of the
adaptation of a propeller to a given airplane and motor.
This question can be well studied by constructing a series
of performance curves for the propeller.
60
80
100 120 140
Velocity, Mi./ hr.
FlG. 36.
Consider the propeller of 8 feet diameter used in the last
section, and let it be acting at the surface of the earth. By
giving to n successively the values 800, 1000, 1200, 1400
revolutions per minute, the thrust and power can be plotted
against forward velocity.
The results for the thrust are shown in Fig. 36. It is seen
that:
i. For a given number of revolutions per minute the thrust
decreases with increasing velocity of translation.
80
THE DYNAMICS OF THE AIRPLANE
2. For a given velocity of translation the thrust increases with
an increasing number of revolutions per second.
These results are to be expected. For a reference to
Fig. 32 shows that for n fixed, increasing V diminishes the
angle of attack of the various elements of the blade, and hence
decreases the resultant air reaction, and consequently the
thrust, since the total reaction is approximately normal to
the chord. On the other hand, for a fixed V, increasing the
value of n will have the opposite effect.
100 120 140
Velocity, Mi,/hr.
FIG. 37.
The curves representing the power are shown in Fig. 37.
It is evident that:
1. For a given velocity of rotation the power absorbed by the
propeller decreases with increasing forward velocity.
2. For a given velocity of advance the power necessary increases
with increasing rotational velocity.
The efficiency can be found from Fig. 35. For convenience
of reference the following table of values for V/nD for the
values of V and n used is given:
THE PROPELLER
81
v^*
s^n
800
1000
1 200
1400
60
-56
45
37
32
70
.66
52
44
37
80
75
.60
50
43
go
-84
67
56
.48
100
94
75
.62
53
no
1.03
.82
.69
59
1 20
1-13
.90
75
.64
73. Motor Diagram. In order completely to solve the
problem of adaptation of the propeller to the machine, it is
necessary to know the manner in which the power of the motor
varies with the number of revolutions. Up to a certain limit
150
cJ
a
75
^-
-^
s.
/
X
\
\
x
X
\
X
x^
\
\
X
X
\
s
600 800 1000 1200 1400 1600 R.P.M.
FIG. 38.
the power of the motor is approximately proportional to the
number of explosions per minute, that is, to the number of
revolutions. After that, due to choking, the power decreases
rapidly with the number of revolutions. A diagram giving a
motor's performance is as shown in Fig. 38. This diagram is
constructed for full admission. When the motor is throttled
a similar curve is obtained, lying below that given.
74. Consider now the conditions under which a given
propelling plant will propel a given airplane. Suppose we wish
the machine to be flown horizontally near the earth at a velocity
of 80 miles an hour. In 22 we described the method of
82 THE DYNAMICS OF THE AIRPLANE
plotting the traction necessary against forward velocity. Sup-
pose the traction is 400 pounds. From the thrust diagram
for the propeller, we find the number of revolutions per minute
that the propeller must be making in order that it shall produce
a thrust of 400 pounds when advancing at 80 miles an hour.
Suppose the value of n thus found is noo. We next turn to
the power diagram and find what power must be furnished to
the propeller. Suppose it is 85 horse-power. We finally
consult the motor diagram and see what power it will furnish
with full admission if running at noo revolutions per minute.
If this is in excess of 85 we can, by throttling the engine properly,
supply to the propeller exactly the power which will make it
turn with the proper number of revolutions that will give
it the forward velocity and traction that are necessary to sustain
the machine.
75. While the curves which have already been given allow
us to answer questions relative to the adaptability of a pro-
peller to a given motor and airplane, it is possible to construct
a diagram which will give a more comprehensive view of the
problem.
On the power curves we construct curves of equal efficiency.
For example, let us construct the curve representing an efficiency
of 60 per cent. For n = Soo } say, we find from Fig. 35 the
values of V. On the power curve for 800 rev./min. we locate
the points corresponding to the values of V. We do this for
n = iooo, 1200, 1400, etc., and through the points found draw
curves. This gives us the 60 per cent efficiency, curves. Simi-
larly, we construct the curves for E = 6$ per cent, 70 per cent,
75 per cent. Assuming that we do not desire any regime of
operation where the efficiency falls below 60 per cent, we have
a diagram which gives clearly the combinations of V and n
which will give us an efficiency of the desired amount.
76. On this diagram we next construct a curve which repre-
sents the functioning of the motor for full admission. To
do this, consider the performance diagram for the motor. For
a given value of n read the power the motor will furnish, and
locate in Fig. 39 on the curve for the same value of n the point
THE PROPELLER
83
where the propeller absorbs that same power. This gives us
a curve which intersects the two limiting efficiency curves
already drawn. Under normal conditions motor and propeller
175
150
125
75
50
25
L
70
.90 100 110 120
Velocity Mi./hr.
FlG. 39.
130
140 150
must be made to operate for values of V and n that lie within
the triangular area which is determined in this way.
77. From the diagram last made it is easy to construct a
diagram that gives the maximum useful power available from
the motor-propeller group. Multiply the value of the power
84
THE DYNAMICS OF THE AIRPLANE
along the full admission curve by the value of the efficiency,
and plot the product against the value of V. This gives the
curve P M shown in Fig. 40. For a given value of V this curve
100
75
P 50
i
ff.
1
A
/2\
Z
/\
A
40
70
100
130
FIG. 40.
gives us the greatest useful power we can obtain from the
propeller attached to the given motor. The efficiency curves
are carried over from the last diagram in an obvious way.
Also curves representing the number of revolutions per minute
are easily constructed.
THE PROPELLER 85
78. To answer the question of the adaptability of the motor
and propeller to the airplane, we plot on the last diagram the
useful power necessary to propel the airplane, as a function
of V (see 24). This gives us the curve P A . We plot only
that amount of it within which operation would be safe. Call
the extremities of the curve a and b. The curve will, in general,
intersect P M in two points a' and b' '. Thus the motor-propeller
with full admission will propel the machine in horizontal flight
at two different velocities. The degree of adaptability of the
motor and propeller to the airplane depends upon the relative
positions of a and a', b and b'. If b is beyond b', as shown,
the motor and propeller are unable to get the speed out of the
machine that its construction would safely allow. If b' were
beyond b, the motor and propeller could develop a velocity that
would be dangerous.
The greatest velocity of ascent will be attained when the
excess power available is a maximum. This corresponds to
a velocity of about 90 miles an hour in the case represented by
the diagram given.
CHAPTER VI
PERFORMANCE
i. CEILING
79. In 48 we have given a discussion of the height to
which a machine can fly. That height, which we called the
ceiling, depends upon the angle of attack. The maximum of
these heights for all angles of attack we called the true ceiling
of the machine. A reference to 44 would lead us to expect
that the true ceiling would be reached when the angle of attack
is the angle of minimum power, for then it would seem that
there would be the greatest surplus power available for climb-
ing. This conclusion depends, however, upon the supposition
of constant available power, a condition, which as indicated
in 44, does not exist. As a matter of fact, the true ceiling is
attained for an angle of attack more nearly equal to the
optimum angle, and sometimes even smaller.* In order to
determine the height of the ceiling by the method given in 48,
it is necessary to make an ascent of some height. We shall
now consider the question of determining the height of the
ceiling directly from the known motive power of the machine,
its fineness, etc.
80. If the ceiling is to be as high as possible for a given
machine and motor, it is necessary that the propeller be well
adapted to the two. For suppose that the machine has risen
to the greatest possible height. The motor will be running
with a certain definite number of revolutions per minute. If
this number is not that which gives the greatest possible power
from the motor, and the value of V/nD is not that which gives
* The Sorbonne lectures of Professor Marchis, Spring semester, 1919. i
PERFORMANCE 87
the greatest efficiency to the propeller, it is evident that there
is a lack of proper adaptation in the various elements of the
machine.
In the calculations that we shall make of the theoretical
ceiling in terms of the motor power, propeller efficiency, and
fineness of the machine, we shall assume that there is a com-
plete adaptation. We shall therefore obtain a quantity which
will in practice exceed the performance of the machine.
Let Po be the power of the motor at the ground, in horse-
power, and \L Z the ratio of the height of the barometer at altitude
z to its height at the surface of the earth.* Then the motor
power at altitude z, in foot-pound-seconds, is
Let E be the propeller efficiency, which, of course, varies
as the machine rises. But in accordance with what was stated
above we shall assume that it has attained its maximum value
at the time the machine ceases to rise. The useful power
available at the ceiling is therefore
The traction necessary for horizontal flight is BW, where
B is the fineness, and W the weight. Therefore the power
necessary is
P m = B'W-V h)
where V* is the horizontal velocity, in feet per second, at the
ceiling.
Consequently, at the ceiling we have
The fineness B is independent of the altitude, but
p*
* The remainder of this section is taken directly from the lectures of Professor
Marchis.
88 THE DYNAMICS OF THE AIRPLANE
where VQ is the horizontal velocity at the surface of the earth,
and p g the ratio of the density of the air at altitude z to the
density at the ground.
Inserting the value of V h and dividing by Po-E, we have
B'W-Vo I
as the relation which must be satisfied at the ceiling.
If we put
, /r _B'W'V
~
we have a number characteristic of the complete machine.
Into it enters the power of the motor, the efficiency of the
propeller, the total weight of the machine, the fineness, as well
as the horizontal velocity at the ground. It would again
seem from this expression that the angle of attack to reach
the ceiling should be that which renders BVo a minimum,
that is, the economical angle. But on account of the con-
siderations adduced in 79, the fineness and velocity for the
optimum angle are used.
If we give to If a succession of values, such as .8, .6, .4,
.3, .2, which will cover those that occur with customary machines,
and for each such value of M, compute the value of the quantity
M
M * 7?'
for a succession of values of z, differing, say, by 1000 feet,
until we reach a value of z for which the quantity in question
is zero, we shall have the values of the ceiling that correspond
to the various values of M. In this way we have a table that
will give the theoretical ceiling for a machine, as soon as we
know its characteristic number.
If, for example, we take M = .4 we find for 2 = 17,000 feet,
M
2 -- = .526-. 4X1.308 = .003.
Therefore we can take 17,000 feet to be the approximate theo-
retical ceiling.
PERFORMANCE 89
Another remark may be added. The weight W includes
the weight of fuel. This continually decreases, so that M
decreases. The ceiling thus continually increases as the time
increases, until such part of the fuel remains as the pilot desires
to have at his disposition in the descent. If we include in W
the weight of the normal fuel carried, the calculation we have
made gives the approximate height to which the machine rises
before its course becomes sensibly horizontal.
81. Supercharge.* If it is desired to make a long flight,
a quantity of fuel in excess of the normal is added. This
increases W and thus M, and therefore lowers the ceiling.
A similar condition exists if any extra load is carried, though
in the first case the surplus load continually disappears, while
in the second instance that is not the case, unless perhaps the
machine is such a machine as a bomber.
We shall investigate the change in the height of the ceiling
produced by a supercharge. At the ceiling we have, as the
equation of power,
The equation of sustentation gives us
W = Ph K v AV J ?.
Let us assume that p/, = ju/ as an approximation.! Dividing
the two equations we find
B'KyA '
If we assume that E is constant, we see that V h is constant.
Therefore the velocity at the ceiling is independent of any
supercharge in weight that the machine may be carrying.
A reference to the equation of power then gives
W
= constant.
* Devillers, loc. cit., pp. 173-177.
f For the accuracy of this approximation see the Appendix, 4.
90 THE DYNAMICS OF THE AIRPLANE
Suppose now that W represents the normal weight of the
machine and h its normal ceiling. Let the machine be super-
charged to a total weight W, and let Jj be the ceiling for this
weight. We have
Therefore
W
As all the quantities on the right are known, this equation will
determine /M> and consequently h f .
For example, suppose a machine with a normal weight of
1500 pounds has a ceiling of 20,000 feet. Let the machine be
supercharged to weigh 2400 pounds. What will be the new
ceiling? We have /** = .468. Hence,
^ = .468 ^ = .748.
1500
The new ceiling is then approximately at 8000 feet.
2. RADIUS OF ACTION *
82. Another problem of interest is that of calculating the
distance to which a machine can fly. In addition to the other
characteristics that must be taken into account we must now
consider the total amount of fuel carried and its rate of con-
sumption. Assumptions must again be made that will render
the results only approximately correct, but nevertheless they
will be of value as a basis of estimates as to performance.
Let P t = power of the motor at time / (/ = o at the start) ;
Q = total weight of fuel, gasoline and oil, at time of
departure;
Qt = weight of fuel at time /;
m = weight of fuel consumed per horse-power per hour ;
W = weight of machine at time of departure;
B = fineness of the machine;
Wt = W Qi = weight of machine at time /.
* Devillers, loc. cit., Chapter XII.
PERFORMANCE 91
Assuming that the rate of consumption of fuel per horse-
power is independent of the altitude, we have
,~ ti.
dQ t = -~ at,
3600
as the consumption in time dt, the second being taken as the
unit.
The useful power is given at any instant by Wt-B-V. In
order to obtain the power in terms of the performance of the
motor we must assume a constant efficiency for the propeller.
We shall take it to be .75.
The useful power developed by the motor in foot-pound-
seconds is therefore .75X550XP/. We consequently have
.75X550 XP=T
Hence,
.75X550
and therefore,
mWl BV
3600 X. 75X55
= mWtBV
1,365,500
Let L represent the horizontal distance traversed. The
machine in reality will be Continually rising, due to decrease
in weight from fuel consumption. However, except at the
start, it will be practically horizontal. We take therefore
dL = Vdt. The relation between the element of path and fuel
consumption is therefore
,r 1,365,500 ^(^1,365,500 dQ t
mB 'W t mB ~'W-Q t '
Integrating, and noting that at t]ie start Q; = o, and at the
instant of arrival Q t = Q, we have
, 1.365.500 , W
T _ '*"* *J ' *J locr
J^t ^rt^ s~\*
mB W Q
It is apparent from this that the greatest distance can be
covered with a given amount of fuel if flight takes place at
92 THE DYNAMICS OF THE AIRPLANE
the angle that gives the smallest value to B. Therefore the
optimum angle should be used. During the flight the machine
consequently flies at its ceiling, 79.
We shall assume #2 = 5.5 and = .12. When we change to
common logarithms and to miles we find:
L-gooo logio pr, approximately.
*~w
From this it is apparent that the distance that a machine
can fly, under the suppositions we have made as to efficiency
of propeller, fineness, and consumption of fuel, depends only
on the ratio of the weight of fuel with which the flight is started,
to the total weight at the time of departure.
In order to cover a great distance with a machine without
replenishment of fuel it is, of course, necessary to greatly super-
charge the machine at the start. This lowers the ceiling, as
was stated in the last section, and it is necessary to know
that the ceiling has not been lowered to such a point that
flight would be dangerous.
Let us assume that a flight of 3000 miles is desired. Hence,
from which
and therefore
Q -u
W~
The weight of fuel carried must then be approximately equal
to the weight of the remainder of the machine.
Suppose the values of m and B for a machine are not those
used to obtain the formula for L. We should then multiply
C j* 12
the value of L given by the formula by X 1 ^-.
m n
PERFORMANCE 93
83. Let us now consider a flight that consists in going a
certain distance and returning to the starting point, the com-
plete flight to consume the entire fuel supply. It is evident
that more fuel will be consumed in the going part of the trip
than in the return. We desire to know the amount of fuel
that can be consumed in the first part of the journey, and leave
assured the possibility of return.
Let WQ be the weight of the machine without any fuel
(dead weight), Q the total weight of fuel at the start, Q g the
part of the fuel that will be expended in going, and Q r the part
expended in returning. The going part of the trip can be thought
of as that of a machine with dead weight of W+Q r and a fuel
load of Q y . The returning trip can be regarded as that of
a machine of dead weight WQ and fuel load Q r . As the distances
traversed are the same we have from the formula of the last
section,
Qr Q*
which can be written
Q-Q,_Q*
W-Q g W
From this we find
Therefore,
14/
since the only admissible root is one less than W.
If we develop the expression for Q 0) we find we can write
Ql^-L. 1 +
Q 28 W
which gives the excess over half the total fuel which can be
consumed before the return trip is started.*
* The method of 83 can be extended to determine the performance of a
bombing plane, or a machine carrying freight. In such cases there is a definite
and considerable decrease of weight at a certain point in the journey, for instance
the place where the return trip starts. See Devillers, loc. cit., pp. 185, 186.
CHAPTER VII
STABILITY AND CONTROLLABILITY
84. We come now to the consideration of a question which
is of the greatest importance in the actual practice of flying,
and of the greatest interest as a problem in dynamics. This
question is that of stability and controllability. We shall
begin the discussion by setting forth some general ideas in^
volved in the problem, so as to make clear exactly what it is
that is desired.
In a broad way the term stability relates to those properties
which an airplane must possess in order that it be " air- worthy/'
that is, that it will be able to fly without too great an element
of insecurity and danger. When we recall the fundamental
principle by which flight is possible, namely, an equilibrium
between tractive force of a propeller, air resistance on the
machine, and gravity, and remember that the air resistance
is delicately dependent upon speed and aspect of the machine
with reference to the wind, while gravity is a force entirely
Beyond our control, we see that the problem will be one into
which a thorough inquiry must be made. Once the equilibrium
between the forces is destroyed, will it be possible to re-establish
it? The air is never absolutely calm; and at different altitudes
and in different localities, eddies and gusts of different natures
and varying magnitudes will be encountered, so that flight
under the ideal conditions that we have thus far assumed will
never exist. Therefore, if secure flight is possible, it will be
accomplished by a more or less frequent recurrence of states
of losing and regaining equilibrium.
85. As the question turns primarily on the air reaction, we
can seek to maintain equilibrium between the forces in two
94
STABILITY AND CONTROLLABILITY 95
general ways. We can construct the machine in such a way
that it will inherently possess properties that will make it
stable, and we can equip it with movable control surfaces that
enable the pilot to alter the air forces and thus assist in re-
establishing the equilibrium. These same control surfaces will
also be the means by which the pilot maintains command of
the machine, directing it along the course that he desires to
follow. Thus we have the two qualities, stability and con-
trollability, both necessary conditions, related and dependent
upon each other, and as we shall see, to a certain extent incom-
patible.
86. In order to be inherently stable a machine must auto-
matically maintain the same attitude towards the relative wind,
for this will tend to keep the air reaction unaltered. When
gusts are met, the machine will thus tend to head into them,
and if this tendency is very strong it may be difficult for the
pilot to keep the machine flying in the course he desires. In a
general way, the greater the inherent stability the more difficult
it will be to make the machine respond to the controls. It
will tend to combat changes in its course. In some instances
a great sensitiveness to controls is a necessity. Thus in the
case of a battle plane, where rapid maneuvering is essential,
the pilot must be able to get a quick and pronounced effect
by moving his controlling surfaces. The degree to which a
machine possesses inherent stability accordingly depends upon
such things as size and purpose. While it is a quality that
all machines must possess to a certain extent, still a condition
can exist that could be described as " too stable." The
machine then could be controlled only at the expense of con-
siderable fatigue to the pilot, and furthermore, it might be
an uncomfortable vehicle, owing to rapid oscillations that
would arise when it encountered gusts.
87. In order to deal with the problem of stability by
mathematical analysis, greater precision must be given to our
definition. In fact, we can distinguish between stability in
the general sense that has been considered, and in the special
restricted mathematical sense to which we proceed, and which
96 THE DYNAMICS OF THE AIRPLANE
will be developed to some extent in the next chapter. There
can easily be a difference of opinion as to the degree in which
it is necessary, or even desirable, that a machine should possess
mathematical stability. On the other hand, it is agreed that
a machine which is mathematically unstable in certain respects
would be a very unsafe and undesirable one.
88. The question of stability is one that can enter into
most dynamical situations. It relates to the effect of a dis-
placement from a position of equilibrium. Such a displacement
gives rise to changes in the forces acting. If the effect is to
restore the state of motion to that which existed prior to the
disturbance, the equilibrium is said to be stable; if the system
tends to depart farther and farther from the prior state, the
equilibrium is said to be unstable; if the system is indifferent
to the change, and maintains a state of motion near to that
which would have existed if the disturbance had not taken
place, the equilibrium is said to be neutral. As an example
of stable and unstable equilibrium, consider the motion of a
marble down a hollow inclined pipe. Let it be on the inside.
It will follow the lowest element, and if, in its motion, it be
slightly deflected, it will, after oscillating back and forth,
regain its former state of motion: its equilibrium is stable.
Suppose, on the other hand, that the marble is on the outside
of the pipe, and rolling down the top, which we shall consider
slightly flattened. A small disturbance will cause it to depart
from its path and fall from the pipe: the equilibrium is
unstable. As an example of neutral equilibrium, let the marble
roll down an inclined plane. A small lateral disturbance will
cause it to follow a course near that it would have followed
had it not been deflected : its equilibrium is neutral.
89. The example of stability which has been given illustrates
two distinct ideas that enter into stability. After the dis-
placement has taken place, there must exist a force tending
to restore the former condition. If such a force exist, there is
said to be static stability. The effect of this force will in
general be such as to make the system with which we are
dealing return to the position of equilibrium, and then depart
STABILITY AND CONTROLLABILITY 97
from it in the opposite sense. In this way oscillations will arise.
If these oscillations die out as the time increases, there is said
to be dynamic stability.
We shall in this chapter give a development of the simpler
aspects of the problem of stability of an airplane, mainly those
connected with static stability. The consideration of dynamic
stability is reserved for the next chapter. A general under-
standing of the means by which stability and controllability
are obtained can be secured by a simple analysis. It is only
by such a preliminary procedure that we can construct models
and make the proper and elaborate experiments that will
furnish the data for an investigation of dynamical stability.
90. In our first analysis we shall consider the effect of
rotation upon an airplane. To distinguish different types of
rotation we draw three axes in the machine meeting at the
center of gravity, one perpendicular to the plane of symmetry,
one in the plane of symmetry, parallel to the propeller axis,
for instance, and the third perpendicular to these two. A
rotation about the first axis is described as pitching, about
the second as rolling, about the third as yawing. We shall
consider the effect on stability of rotations of these three
types. The general rotation that a machine can experience is
a combination of all three. The more elaborate investigation
in the next chapter covers that case. It is, however, by con-
sidering separately the three types of rotation that one is led
to develop principles of design that make stability possible.
We shall first consider the effect of pitching, which gives
rise in its simplest form to what is called the problem of longi-
tudinal stability.
LONGITUDINAL STABILITY
91. We can say in an approximate way that longitudinal
stability depends primarily on the manner in which the center
of pressure moves as the angle of attack varies. Consider a
plane surface. Suppose it were the sustaining member of a
machine. In horizontal flight the resistance R, the traction J 1 ,
98
THE DYNAMICS OF THE AIRPLANE
FlG. 41.
and the weight W meet at a point, say, the center of gravity.
Suppose some influence tended to rotate the machine in such
a way as to increase the angle of
attack. The resistance R recedes from
the attacking edge, remaining sensibly
parallel. A moment then arises that
tends to return the machine to its origi-
nal position. It is therefore statically
stable.
Suppose now that we had for sus-
taining member a single cambered sur-
face. With increasing angle of attack
the center of pressure tends, as a rule,
to approach the edge of attack, and
vice versa. Thus the air pressure tends
to create a couple that will increase the
displacement, and we would expect the machine to be static-
ally unstable.
92. An analysis of this sort should be pursued further.
Consider the surfaces of a biplane. For different angles of attack
suppose that the line of action of the result-
ant air pressure were drawn in the plane of
symmetry. This gives a one-parameter family
of straight lines. They envelop a curve, called
the metacentric curve. For the angles of
attack that are used, this curve is concave
to the front, as shown in the figure. The
point of tangency of the metacentric curve
and the line representing the air reaction is
called the metacenter. As the angle of attack
increases, the metacenter describes the meta-
centric curve in the sense A to B.
Suppose the machine is in horizontal flight.
Let the propeller pass through the center of gravity G; then the
machine being in equilibrium, the air reaction R must also pass
through G. Suppose the machine pitches so as to increase the
angle of attack. The metacenter moves to m', the line of action of
R'
STABILITY AND CONTROLLABILITY 99
the air reaction to R'. A couple of moment R'xGP is created,
where P is the foot of the perpendicular from G on R f , and R'
represents the magnitude of R' . This will be a restoring
couple if G is below K, the intersection of R and R'\ other-
wise, it will tend to increase the displacement. Therefore there
will be a condition of equilibrium if G is below the metacentric
curve. The metacentric curve, however, in general, lies too
low for this condition to exist. Stability must therefore be
obtained by some added feature. This feature is a tail plane.
93. Tail Plane. Consider a plane surface to the rear of
the sustaining surface. Let ab be a section of it. Suppose
that the air reaction on it is originally zero, that is, that it is
in the bed of the wind. Let G be the center of gravity.
Suppose the machine is rotated about the center of gravity
FIG. 43.
through an angle 0; then the tail plane takes a position a'b',
inclined at an angle 6 to the relative wind. The air reaction, /,
on it will be practically normal to it, and can approximately
be represented by ksdV 2 , where k is the constant given in 3,
and 5 is the area of the tail plane. There thus arises a restoring
moment of amount
where d is the distance shown in the figure. The air reaction
on the main sustaining surface likewise has a moment, about G.
The sum of the two must be a restoring moment, in order that
the tail plane function in the desired manner. Similar con-
siderations must hold for a motion that tends to decrease the
angle of attack.
It is seen at once that a problem of great importance is the
determination of the proper area for the tail plane, and its
100 THE DYNAMICS OF THE AIRPLANE
distance from the sustaining surfaces. The angle at which it
is set is also important. It is found that it must make a
smaller angle with the relative wind than the main plane.*
The tail plane does not act as an independent plane, but is
greatly influenced by the wash of the main planes. Its actual
behavior must be determined by experiment on different
combinations. Investigations of this sort have been conducted
by Eiffel, who has found that the wash from the front surface
has the same effect as decreasing the angle of incidence of the
tail plane. Thus, if the tail plane is apparently in the bed
of the wind, it is in effect behaving as though it were at a nega-
tive incidence, and therefore has a downward pressure exerted
upon it.
94. The Elevator. For allowing control of the machine,
and increasing longitudinal stability, it is fitted with an elevator.
This is the plane attached to the rear of the tail plane, and
FIG. 44.
movable about a horizontal axis. By means of it the angle
of attack of the machine is altered. Suppose the machine is
in horizontal flight with the elevator in the neutral position
ab. Let it then be turned through the angle 6 into the posi-
tion be. There arises a force /, which we can represent with
sufficient accuracy by
, , /T70 sin 6
r '=ks'V 2
.4 +.6 sin 0'
where k is a constant and s' the surface of the elevator. Suppose
the center of gravity is at G'. Let Gb = d\ then assuming the
* Bothezat, "fitude de la Stability de 1'aeroplane," p. 98.
STABILITY AND CONTROLLABILITY
101
force r r as normal to the elevator, we have for the moment
created :
, , k , T79J sin 26
M = -sV 2 d- .
2 .4+. 6 sin
The variation of this force with the value of is shown by a
consideration of the function
sin 20
4-f-.6 sin
This function reaches a maximum between 35 and 40.
It therefore follows that an elevator should not be given an
inclination to exceed something like 30.
The moment caused by turning the elevator into the
position shown in the figure will have the effect of rotating
the machine in such a direction as to increase the angle of
attack. In the meantime the moment of the air reaction on
the sustaining surfaces changes. The machine will rotate to a
position such that the moment of all forces (propeller traction,
air reaction on sustaining members, tail plane, elevator) passes
through the center of gravity. If the motor power is at the
same time properly altered, the machine will fly at a different
angle of attack.
The means for changing the position of the elevator are
secured by equipping it with a lever arm perpendicular to its
surface, to the end of which
a wire is attached, which in
turn runs to the cock-pit.
To turn the elevator may
require considerable muscu-
lar effort on the part of
the pilot. This effort can
be lessened by properly bal-
ancing the elevator. The
figure shows schematically
the method of doing this.
Let aABb be the fixed tail
plane, and ABcde the elevator, pivoted about AB. It is
d
FIG. 45.
102 THE DYNAMICS OF THE AIRPLANE
obvious that, by this disposition of the axis, the air reaction
can be kept fairly near the. axis so that the moment required
to turn the elevator will be lessened. The resultant air reaction
must not, however, in any position of the elevator pass through
AB, for the pilot must always be able to feel a tautness in the
controls.*
95. It is seen that we have considered only the question
of static longitudinal stability, that is, the question of the
existence of a couple tending to return the machine to its
original position. Granting that such a couple exists we cannot
ascertain the ultimate effect of the oscillations it will produce
without knowing its magnitude as related to the angle through
which the machine has been turned. We see also that while
the rotation is going on, the instantaneous angle of attack of
every element of the wing is changing, and the change of air
reaction arising in this way is evidently dependent upon the
velocity of rotation. It is possible to analyze with some
approximations these various agencies that are at work, and
obtain a differential equation from which we can draw con-
clusions as to the dynamic stability.! We shall not do this,
however, for pitching is unavoidably connected with a change
in the motion of the center of gravity, and we defer the whole
question to the more accurate and complete discussion in the
next chapter.
STABILITY IN ROLLING
96. Suppose the machine possessed a single sustaining mem-
ber whose leading and trailing edges were straight lines, that its
body were that of a solid of revolution, that its tail plane were
in the continuation of the axis of the body, that landing gear
struts, wires, etc., were non-existent. Suppose that while
in rectilinear flight with the axis of its body horizontal, it were
made to roll about that axis. That half of the wing whose motion
was downwards, would have its angle of attack increased; the
half of the wing moving upwards, would have its angle of attack
* For a further discussion of this see Devillers, Chapter XIII.
t Devillers, Chapter XIII.
STABILITY AND CONTROLLABILITY 103
decreased. Consequently there would be a damping of the
motion, dependent upon the velocity of rotation. Tail plane
and rudder would also assist in this. When the motion had
died out there would, however, be no restoring couple; that is,
there is no static couple tending to restore the machine. It is
obvious that during the motion, and in the displaced position,
with the wings no longer horizontal, the vertical component
of the lift no longer has its original value, and unless the proper
variation of speed accompanied the process the altitude of
the machine would change. But we are not here concerned
with all the complications that arise. We merely are interested
in seeing that there is no restoring couple.
97. While the discussion that has been given does not
apply in toto to an actual machine,
the general characterization can
be carried over, and we see that
we must provide a means of creat-
ing a restoring couple.
One way of causing stability
in rolling is to set the wings at
an angle, as shown in the figure.
They are then said to possess a
dihedral. The complete action of this is difficult to trace,
but we can note one effect. For equilibrium we would have
zR v cos a = W.
Now let the machine rotate through an angle 6 so as to lower
the left wing. The vertical component of the air reaction
would be, assuming R has not materially changed,
R v COS (a 0) +R y COS (-f- 0) = iR y COS a - COS 0.
This is smaller than it was before. Such a rotation would
then be accompanied by a downward motion of the machine.
This would increase the lift on the left (lower) wing, more than
on the right, and consequently a restoring moment would be
created, that would make the machine roll back to its original
position. The dihedral must not be too pronounced, or diffi-
culty will arise from the lateral effect of a wind.
104 THE DYNAMICS OF THE AIRPLANE
The fact that the dihedral produced stability through the
consideration that rolling will produce a downward motion
shows how impossible it is to separate completely different
types of motion in the discussion of stability. They are all
closely connected, and rotation about one line will cause rota-
tions about other lines, though perhaps of a less pronounced
nature.
98. Controllability in the lateral sense is furnished by
means of the ailerons. Rectangular portions are cut from the
corners on the trailing edge of the wings, and are then pivoted
along their front edges. They are so fastened to the controls
FIG. 47.
that the raising of the ailerons on one side is accompanied by a
lowering of those on the other. Their action is obvious. They
merely increase the lift (and the drag) on one side and decrease
it on the other. This gives the pilot the power of creating at
will a moment that tends to roll the machine.
LATERAL STABILITY
99. Lateral stability is secured by means of a fin, in the
plane of symmetry, to the rear of the center of gravity. The
general manner in which it functions is obvious. In case the
machine is made to yaw, a moment is created tending to restore
the machine. Of course, all parts of the machine have an
effect in producing the restoring couple. Those parts well
forward, such as the housing of the motor, tend to aggravate
the yawing.
Controllability in direction is secured by means of a rudder,
usually placed at the rear of the fin. In order to decrease
the muscular exertion that the pilot must use to turn the
rudder, it is generally balanced, as was explained in the case
of the elevator. Here again it is necessary to be assured that
STABILITY AND CONTROLLABILITY 105
the force on the rudder will not pass through the axis, as in
this case the pilot would not be sensible of any pressure on the
controls. When the rudder is in the neutral position, it acts
as a portion of the fin.
100. The complete analysis of the action of the rudder is
difficult, for its turning introduces a sequence of phenomena
hard to follow. The case is much more complicated than that
of a ship, where the sustentation is in no way dependent upon
speed.
Imagine that the rudder has been turned to the right. A
force / arises on the rudder, producing a moment tending to
make the machine yaw to the right. To find the effect on the
motion of the center of gravity we must apply this force at
that point. We see then that the center < / ^~v
of gravity moves slightly to the left at I A
first. But the yawing produces a force F
on the opposite face of the fin. The effect
of this transferred to the center of grav-
ity will be to make the machine turn to
the right. The total force tending to
make the center of gravity describe a
curve to the right is approximately Ff.
The total moment tending to produce rotation about a vertical
axis is the resultant of the moments produced by the rudder
and fin. It will be seen that a large fin well forward will make
possible a considerable turning force by the use of a fairly
small rudder.
The combined use of ailerons and rudder in turning has
already been discussed in 53.
101. It is not possible to separate a tendency to yaw from a
tendency to roll, for the two will occur together. Thus, if the
machine should start to yaw to the right, the left side of the wing
will have a greater velocity than the right, and consequently
a greater lift, so that the machine will start to roll.
The fin, while designed primarily for producing stability in
direction, also contributes to stability in rolling, if properly
placed. Suppose the machine had no dihedral, and that it
106 THE DYNAMICS OF THE AIRPLANE
started to roll in the direction that causes the left wing to
lower. The lift is no longer vertical, so the machine will start
downwards. Suppose the machine had a vertical fin placed
above the center of gravity. It will have been displaced from
the vertical plane in the rolling, and when a tendency to settle
begins, it is seen that a force arises on the fin that tends to make
the machine roll back toward its original position. If the fin
had been below the center of gravity, the effect on it would be
to accentuate the roll. The efficiency of the fin in creating
stability in rolling will also depend on its distance to the front
or rear of the center of gravity.
In a similar way the dihedral of a machine adds to direc-
tional stability. For if the machine starts to yaw to the right,
the effective angle of attack of the left wing is increased, that
of the right wing decreased, and therefore a righting moment
arises.
From the two properties that have been deduced* it appears
that a dihedral in the wings is equivalent to a certain fin, and
can for practical purposes be so regarded.
102. Spiral Instability. A type of instability that is likely
to occur, and is apt to prove very dangerous, is called spiral
instability. It can be caused by large fin surfaces too far to
the rear of the center of gravity. Suppose a machine with
such a fin were banked so as to turn to the left. As the turn
commences a pressure arises on the left side of the fin. This
causes the machine to rotate so as to keep its axis along the
path. The outside wing, moving faster than the inside, the
lift is greater there, and this tends to increase the banking.
Along with this goes a slipping towards the inside ; for a shifting
in the direction of the lift decreases the vertical force on the
machine. This slipping causes a force on the fin, that tends
to make the movement more pronounced. As a result of this
sequence of phenomena the machine gets into a spin, or rapid
nose dive, from which the pilot may be unable to extricate it.
CHAPTER VIII
STABILITY (Continued)
103. A GENERAL discussion of the question of stability of
an airplane can be made by the methods developed by Routh
in his prize essay on the stability of a dynamical system.*
The application of these methods to aviation was first given
by Bryan, f His results are general, and the question of the
stability of a machine for any slight disturbance from a position
of equilibrium depends upon the nature of the roots of tv/o
biquadratic equations. The coefficients in these equations
depend upon the construction of the machine. All elements
that enter into the machine's construction, the wings, the stabil-
izing surfaces, the controlling surfaces, contribute to the value
of the coefficients. In the application of the methods, difficulties
arise in the determination of these coefficients. Bryan himself
applies the method to machines of certain general characteristics,
for which he can obtain with considerable certainty, values
for the coefficients in terms of wing area, angle of attack, etc.
Extensions and application of the method, by obtaining the
values of the coefficients in the biquadratics through experi-
ments on models in wind tunnels, has recently been made by
other investigators, t
104. Moving Axes. Many problems in rigid dynamics are
best treated by means of moving axes. These axes are fixed
* E. J. Routh, "Stability in Motion," London, Macmillan Co., 1877. See
also his "Advanced Rigid Dynamics," Chap. VI.
t G. H. Bryan, "Stability in Aviation," Macmillan Co., 1911.
t Bairstow, "Technical Report of the (British) Advisory Committee for
Aeronautics, for 1912-13," London, Darling & Son. In this report Bairstow
gives simplifications in the equations of Bryan, and develops a method applicable
to a machine without special hypothesis as to its construction.
107
108 THE DYNAMICS OF THE AIRPLANE
in the body, move along with it, and assume continually
changing positions and directions. They can afford us only
indirectly a knowledge of how far the body has moved, but
are peculiarly adapted to reveal the oscillations, and small
changes in motion, that the body is experiencing at any instant.
And in the question of stability it is exactly such points that are
at issue. Use of moving axes was first made by Euler, and
equations of motion for such axes were obtained by him. Bryan,
however, found the particular system of coordinates used by
Euler not well adapted to follow the motion of an airplane,
and introduced slight changes in them.
105. Suppose that in a rigid body three axes be chosen, with
their origin at the center of gravity. As the body moves,
taking this system of axes with it, we fix our attention upon
it at some particular instant. Its center of gravity has a
vector velocity V, and the body a rotational velocity repre-
sented by another vector R. Resolve V along the three direc-
tions in space that are occupied at that instant by the moving
axes: let the components be u, v, w. Likewise resolve R:
let its components be p, q, r. We thus obtain six functions
of the time, and the equations of motion will be obtained by
properly expressing these six functions and their derivatives
with regard to the time in terms of the forces and moments
acting on the body. Something as to the significance of the
derivatives can be obtained by considering one of them, for
instance du/dt. It measures the rate at which the component
of a vector along a varying direction is changing. It is there-
fore not a component of the acceleration of the body, for a
component of an acceleration is the rate at which the com-
ponent of the velocity along some fixed direction is changing.
Suppose that at instant / the origin is at O, and let the position
of the moving axis be X. Let V be the vector velocity; then u
is the projection of V on X. At a subsequent instant let the
origin be 0' and the x axis be denoted by X' } and the velocity
by V'\ then u' is the projection of V on X'. Thus the incre-
ment of u is u' u. On the other hand, to find the acceleration
of the body at the instant / we must project V upon X, instead
STABILITY
109
of on the new position of that axis. The acceleration along the
fixed direction in space, which at any instant coincides with the
position of the moving axis, can, however, be expressed in
terms of du/dt, v and w. Similar remarks apply to the other
quantities. We shall not derive these forms. They can be
found treated by Routh.*
106. Choose for #-axis a line in the plane of symmetry,
for instance, parallel to the propeller axis, directed backward;
FIG. 49.
for 3/-axis, a line perpendicular to the plane of symmetry to the
left as seen by the pilot; for z-axis, a perpendicular to these,
in the plane of symmetry, directed upwards. The axes are
further so chosen that the origin is the center of gravity of the
machine.
* " Elementary Rigid Dynamics," Chapter V, "Advanced Rigid Dynamics,"
Chapter I.
110 THE DYNAMICS OF THE AIRPLANE
With this choice of axes the accelerations of the center of
gravity along fixed directions in space, that are the instan-
taneous positions of the moving axes are, respectively:
du .
+wq-vr,
dv ,
+ur-wp,
dw . *
+vp-uq.*
We need also the rates of change of the angular momenta
about the various axes. We denote these momenta by hi, fe,
hz'j then
hi=pA-qF-rE,
h< 2 =qB-rD-pF,
hz = rC-pE-qD,
where A , B, C are the moments of inertia, and D, E, F the prod-
ucts of inertia. The machine being symmetric we have
D = F = o. The rates of change of the angular momenta are
then:
i . j ,
-ph-3+rhi,
107. It is necessary for us to have a means of comparing
the orientation of the machine with some fixed orientation.
This standard of reference we take as one in which the xy
*When comparison is made with Routh, "Advanced Rigid Dynamics," 5,
it will be found that the signs of the second and third terms are changed. This
comes from the fact that Routh uses a right-hand system of axes while we have
a left-hand system. In our system a rotation of an ordinary screw from the
#-axis into the y-axis would advance the screw along the negative z-axis.
f Routh, "Advanced Rigid Dynamics, " 5.
STABILITY
111
plane is horizontal. From this position rotate the machine
first about the z-axis through an angle $, then about the ;y-axis
through an angle 6, and finally about the #-axis through an
*1
angle . A rotation about the y \ axis is positive when it turns
z\
the z
x
axis into the x \ axis. The original positions of the axes
are shown by the letters with subscripts o, and the posi-
y l and y<
FIG. 50.
tion after the first and second rotations are shown respect-
ively by subscripts i, 2, and the final positions by the
letters without subscripts. Further, any position of the coor-
dinate axes can be obtained by a unique rotation of the sort
described. Consider a position, represented by x, y, z. The
angle between the planes xz and XQZQ determines ^. Let the
plane XZQ intersect the xoyo plane in Ox\] then 6 is the angle
between Ox and Oxi. Let Oyi be the intersection of the xoyo
and yz planes; then < is the angle between Oy\ and Oy. To fix
112 THE DYNAMICS OF THE AIRPLANE
the orientation of the machine at any instant we draw through
the center of gravity axes parallel to those of reference, and
determine \f/, 6, as just described.
108. Suppose the orientation of the machine is continually
changing; then $, 0, are functions of the time. From their
instantaneous values, and those of their derivatives with
regard to the time, we can obtain the instantaneous values of
pj q, r. We shall be in need of these relations, and will proceed
to their determination.
About the origin in Fig. 50 draw a sphere of unit radius.
Consider the point C. As the x, y, z axes move, C has a velocity
dd/dt along the arc XZQ, and a velocity cos 6d\l//dt normal to the
plane xOzo, that is, along xy. Now the angular velocity r is
the velocity with which x approaches y, that is, the velocity
along xy. But xy makes an angle $ with xy\. Resolving
in the direction xy the velocity dd/dt along ZQX, and cos 6d\l//dt
along xyi, we have
r= sin 0- hcos -cos6~.
at at
Likewise q is the velocity of C along zx\ hence
A de . . d$
= cos - hsm 0-cos 8~.
at at
Finally, r is the velocity of z away from y along yz. But the
velocity of z relative to Co is d/dt, and that of Co itself is
sin 6 - d^/dt. Hence taking account of the positive directions
for increasing \J/ and 4>,
When the angles ^, 6, 4> are all zero, we have
d de d\j/
*The values of p, q, r are similar to the well-known Euler geometrical
equations. See Routh, "Elementary Rigid Dynamics," 256.
STABILITY 113
and when the angles are small, we can also use these values
for p, q, r, with a sufficient degree of precision. The use
of the exact values would introduce a very high degree of com-
plexity into our work.
109. Equations of Motion. The equations of motion are
easily derived from the expressions that have been given in
106. The motion of the center of gravity is determined by
equating the expressions for the linear acceleration to the forces
per unit mass acting along the respective axes. The rotation
will be determined by equating the rates of change of angular
momenta to the moments of the forces about the respective
axes.
The forces acting come from three sources: gravity, pro-
peller traction, and air reaction. The components along the
axes of the force of gravity depend upon the orientation of
the machine. Let ^, 6, be the values of the angles giving
the orientation, as explained in 107. Then the components of
the weight per unit mass are:
g-sin0, g-cos 0-sin , g-cos 0-cos <,
along the x, y t z axes, respectively.
The moments due to the weight are zero.
Next, consider the traction of the propeller. Let it be
parallel to the x-axis, of numerical value H per unit mass, and
applied at a distance h above the x-axis. Its components are:
-H, o, o.
The moments of this force are:
o, -hH, o,.
about the x, y, z axes, respectively.
There remains the air reaction. It depends upon the
instantaneous velocity, translational and rotational; that is,
upon u, v, Wj pj q, r. No simple expression can be given for it.
We denote the components of this force, per unit mass, at a
given instant by X, F, Z, and its moments about the respective
axes by L, M, N.
114 THE DYNAMICS OF THE AIRPLANE
The equations of motion now become
+wq- w =g-sin e+X-H,
at
dv
- h ur wp g-cos 0-sin 0+F,
for the motion of the center of gravity, and (by making use
of the angular momenta in terms of the angular velocities,
remembering that the products of inertia D and F are zero),
B
for the rotation about the axes.
110. We shall employ the equations of motion to determine
the effect of a disturbance that a machine might undergo while
it is in a state of steady motion. We limit ourselves to the
simplest case, for even in this case the equations with which
we must deal assume a sufficiently complicated form. The
analysis of the subject which has been developed to date, will
not enable us to treat a general displacement from a general
position of equilibrium in which a machine might find itself.
We can study some simple cases, those applying to normal
states of flight. If we find in these instances a high degree
of equilibrium, we will be led to infer that the machine can be
regarded as air-worthy, and will furnish a means of transit in
general of a sufficient degree of security.
Suppose the machine is in rectilinear horizontal flight.
The xz plane is vertical. Assume also that the #-axis is hori-
zontal. (We have assumed the #-axis parallel to the pro-
STABILITY 115
peller axis. The new assumption is really not a restriction,
as the equations that follow could easily be modified in such
a way as to take care of the case where in the horizontal rec-
tilinear flight the -axis is inclined at a certain angle. There
would, in general, be some angle of attack for the wings for
which the #-axis is horizontal.) In this state of steady
motion, v, w, p, q. r are all zero, while u has a constant value U
(which is negative, on account of the direction in which the
#-axis is chosen). The steady motion is represented by the
equations:
o = X -H ,
o=Y ,
o = Z ,
o = mLo,
o = mNo,
where a letter with subscript zero denotes the value of that
quantity for the state of steady motion.
Now imagine a disturbance to take place. Its exact nature
we do not specify, nor do we know exactly what effect it will
instantaneously have upon the machine. We shall merely
say that it has altered all the velocities of the machine. Thus
v, w, p, q, r have values other than zero, while u takes the value
U+u. (The change in the significance of u will cause us no
confusion.) As is usual in the case of such discussions, we
assume that u, v, w, p, q, r are small enough that their squares
and products can be neglected.
111. The orientation of the machine will be different. We
take the orientation prior to the disturbance as the one of
reference. After the disturbance the position of the machine
will be represented by ^ ; 6, <, all of which we assume
sufficiently small for us to write
d dB dty
116 THE DYNAMICS OF THE AIRPLANE
The disturbance will have changed all the components of
the air reaction. Consider one of them, X. Putting in evi-
dence the quantities upon which it depends we obtain
X(U+u, v, w, pi q, r) = X(U t o, o, o, o, o)
M +? M + ^. '
dp dq dr
That is,
X = Xo+uXu+vX 9 +wX u +pX 9 +qX q +rX r ,
where XQ is the value of X prior to the disturbance, and X u
X v , . . . Xr are constants called " resistance derivatives," by
Bryan. In all, we shall have thirty-six resistance derivatives,
namely:
5,, S = X, Y, Z, L, M, N,
s = u, v, w y p, q, r.
On account of the symmetry of the machine eighteen of the
resistance derivatives vanish. Those that vanish are
s = v,p,r,
and
&, S=Y,L,N,
To show that these eighteen resistance derivatives are zero,
consider one of them, for instance, X v . If it were not zero,
a sideways velocity v, would cause an increase vX v in the force
along the s-axis, and this would be positive or negative accord-
ing as the displacement were towards the left or right, which
could not be the case if the machine were symmetrical.
Consider also Y q . If it were different from zero, a dip
in the machine would produce a sideways force of a different
direction from that produced by a tip.
Finally, let H +5H be the new propeller traction.
STABILITY 117
The differential equations that represent the motion sub-
sequent to the disturbance will be obtained by substituting the
new values of the velocities in the six equations at the end of
109, dropping products of small quantities, using the new
values for the various forces and moments, and at the same
time making use of the relations for steady motion.
We obtain thus:
~
d*w
B -2 = m(uM u +wM w +qM q h> 5H),
at at
These six equations divide into two groups. The first,
third, and fifth contain only u. w, q, and their derivatives; the
second, fourth and sixth contain v, p, r and their derivatives.
The first group determines motion in the xy place. Such
motions are called symmetric or longitudinal oscillations. The
second group determines oscillations that are called by Bryan
the asymmetric oscillations.
112. As the equations contain also and 0, we replace
p and q by d/dt and dO/dt, respectively, and thus take u, v t
Wj <, 0, r as the quantities to be sought.
Finally, we take 6/7 = o; that is, we assume that the rota-
tion of the propeller changes in such a way that the traction is
unaltered by the disturbance.*
* For discussion of the case where this assumption is not made see Bryan,
loc. cit, p. 28.
118 THE DYNAMICS OF THE AIRPLANE
We make these substitutions, represent differentiation by
the symbol D, and divide the equations into the two groups
mentioned. For the longitudinal motion we have:
(D-X u )u-X w w-(X g D+g)=o,
where we have put k B 2 = B/m; and for the asymmetric oscilla-
tions we have:
(D-Y v )v-(Y p D-g) + (U-Yr)r = o,
A 2 D 2 -L p D)<}>-(k E 2 D+L r )r = o,
where k A 2 = A/m } k E 2 = E/m, k c 2 = C/m.
These are the fundamental equations with which we have
to deal. If we had assumed that the steady motion had been
along a line making a constant angle with the horizon, slightly
different equations would be obtained.*
113. The equations that we have to solve are linear equa-
tions with constant coefficients. The solution is most readily
accomplished by means of symbolic operators, f Consider first
the equations for longitudinal motion. Let
-Xu, -X w -(X a D+g)
-Z u , D-Z W , -(Z Q +U)D
-M u , -M w , (k B 2 D 2 -M q D)
The determinant on the right is to be developed according to
the ordinary method, the symbol D representing differentiation.
When this is done we find
* Bairstow, loc. cit., Bryan, Chapter I, Cowley and Evans, Chapter XI.
t Murray's " Differential Equations," Chapter XI. Wilson's "Advanced
Calculus," p. 223.
STABILITY 119
where A, B, C, D, E are constants, whose explicit values will
be given presently. The quantities . w, 6 are then all solu-
tions of the single equation of the fourth order,
In a similar way we put
A '-ir rr -( Yp D-g), U-Yr,
-_.,. (k A 2 D 2 -L P D) -(k E 2 D+L r ) ,
-N,, -(k E 2 D 2 +N P D), (kc 2 D-Nr)
which gives, upon development,
The quantities v, 0, r are then all solutions of
The conditions for stability can now be given in a general
way. The quantities u, w, will be linear combinations of
the form
where Xi, X?, Xa, X4 are the four roots of the quartic
and v, , r will be of the same form, where Xi, X2, Xs, X4 are
roots of
In order that the machine be stable it is therefore necessary
that the roots of the two quartic equations have their real parts
negative. For if this condition is fulfilled the values of u,
v, w. 9, 0, r become smaller and smaller as the time increases;
they thus approach zero, and a steady motion is therefore
resumed. If, on the other hand, the real part of one of the
roots is positive, the term corresponding to it increases and
instability will result. It is to be noted that a machine can
have longitudinal stability, but asymmetric instability, or
vice versa. (We shall, however, see later that asymmetric
stability is the more difficult to obtain.)
120
THE DYNAMICS OF THE AIRPLANE
114. The necessary condition that the real roots be negative,
and the real parts of the imaginary roots be negative in a
quartic equation is given by Routh.* Let the equation be
Then the roots will have the property mentioned if, and only
if, the coefficients
a, b, c, d, e,
and the discriminant, called Routh's discriminant,
bcd-ad 2 -eb 2
are all positive,
An airplane will therefore be stable longitudinally if
A, B, C, D, E, and BCD-AD 2 -EB 2
are all positive.
It will be asymmetrically stable if
A,, B lt Ci, Di, 1, and B.C^-A^-E^
are all positive.
115. If the determinant A be developed, we find:
A = kJ, B=-(M q +X u k B 2 +Z w k 2 ),
z w .
M w ,
X V) X w
7 7
Ss U . / s ir
D=-
f u , M WJ M q
E=-g Z M , Z w .
In the same way, by developing A' we obtain:
^i =
k 2
* "Stability in Motion," p. 14.
STABILITY
-kj, -kj
k E
2 , Lr
-L P , -
-k F ? ,
he 2 , k c 2
2 , Nr
N P ,
kc 2
'., Y p , o
- F., o,
Yr-U -
\- L p , Lr
c, Lpj K E
L v , -k A 2 ,
L r
N P) Nr
r ,, N P) k E 2
N V) k E 2 ,
Nr
V V V 77
JL V) * p, * r U
+g
L,, -k E 2 ,
L c , L p . Lr
N ,
kc 2
121
*, Lr .
116. The whole question of determining the stability of a
machine depends upon the determination of the resistance
derivatives, the knowledge of which will give the coefficients
in the two quartics. This is a practical question which is
solved by experiments on a model in a wind tunnel. We shall
give only a slight indication of the procedure. The question
is discussed at length by Bairstow.*
Consider the resistance derivatives that depend on u.
By means of instruments which hold the model in the wind
tunnel, the resistance XQ for the velocity U is determined, the
#-axis being in the direction of the wind. We assume
X = cn\
where c is a constant and u the velocity. Hence
(}X -yr XQ XQ
== .A. == 2 CU == 2 U ' ~ =: 2 .
* Loc. cit., pp. 154-158. See also: Hunsaker, "Experimental Analysis of
Inherent Longitudinal Stability for a Typical Biplane." First Annual Report
of the (American) National Advisory Committee for Aeronautics, Hunsaker,
"Dynamical Stability of Aeroplanes," Smithsonian Miscellaneous Collection,
Vol. 62, No. 5. These two papers by Hunsaker will be referred to as Hunsaker
(i), Hunsaker (2), respectively. The second paper especially gives a full dis-
cussion of the subject referred to here.
122 THE DYNAMICS OF THE AIRPLANE
In the same way Z, Af, N u are found from a knowledge of
ZQ, MQ.
The derivatives that depend upon v are found by turning
the model about the s-axis through some fixed angle, for the
effect of such a turning is to give a component of the air reaction
parallel to the ^-axis. In the same way the derivatives depend-
ent on w are found by making measurements after the machine
is turned about the y-axis through some angle.
The derivatives that depend upon p, q, r are termed rotary
derivatives by Bryan. They are determined by oscillating the
machine about the various axes, controlling the oscillations by
means of springs, and measuring the damping by means of
photography.
A discussion of the range in numerical value that may be
expected in the resistance derivatives for machines generally
in use will also be found in the report given by Bairstow.
117. As an example of the numerical values, we take those
given by Bairstow for a typical machine. They are:
k B 2 = 25 m= 40 slugs*
X u =-
.14
Z u =- .80
Mu =
x w =
.19
Z w = 2.89
M w =
2.66
x q =
5
Z 3 = 9
M q =
210
k A 2 =
25
kc 2 = 35
W*
o
Y v =-
2 5
L v = .83
N v =
54
Y p =
i
L p = 200
N> =
28
Yr=~
3
Lr= 65
Nr =
37
The equation for longitudinal stability is
and for lateral stability
X 4 +9.3iX 3 +9.8iX 2 -f-io.i5X-.i6i=o.
"The engineering unit of mass, equal to 32.16 pounds. It is used here in
order to have consistent units. See appendix for a further discussion.
STABILITY 123
The Routh discriminant for the first equation is found to be
positive. Therefore the machine possesses longitudinal stability.
The negative term in the second equation shows the machine
to be asymmetrically unstable. Disturbances therefore would
need to be corrected by the assistance of the controls. Bairstow
shows that the machine becomes asymmetrically stable when
gliding at a glide of i :6 with the propeller cut off. To prove this
it is merely necessary to have assumed that the steady motion
had been in a line inclined at some angle to the horizon. The
development is similar to that which has been given here.
118. The stability of a machine will depend upon its speed,
and a machine may be stable at one speed and unstable at another.
This is due to variation of the various resistance derivatives.
Thus, while it is possible to make a machine with a considerable
speed range through a change of elevator setting, the machine
cannot be expected to be uniformly stable at all the speeds
at which it may fly. At high speed the pilot may be able to
abandon his controls, the machine possessing so great an
inherent stability as to be able to "fly itself," while at low
speeds constant watchfulness and attention may be necessary.
An interesting example of this sort is given by Hunsaker.* In
his report the longitudinal stability of a machine is investigated
for speeds varying from 79 miles an hour, corresponding to an
angle of attack of i, to a speed of 43.7 miles an hour, corre-
sponding to an angle of attack of 15. 5. Instability at low
speeds corresponding to an angle of attack larger than io.5
results from the Routh discriminant becoming negative.!
119. The general discussion of the quartic equations under
consideration presents considerable difficulty. Bairstow has
given an approximate factorization of the equations.! That is,
he has given factors that, while not algebraically exact, give
a fairly close approximation when the numerical values of the
coefficients are used. And from the factors he gives a general
* (i) Loc. cit., pp. 47-51.
t Hunsaker (2), p. 65, considers the lateral stability of a Clark tractor and
shows it is laterally stable except at low speed.
I Loc. cit.
124 THE DYNAMICS OF THE AIRPLANE
idea of the properties of the plane that contribute to its sta-
bility.
The equation for longitudinal stability Bairstow factors into
In general, it may be said that the first factor represents a
very short oscillation, which in the majority of machines will
die out quite rapidly. The second factor, however, represents
a relatively long oscillation, which is dependent in its nature
upon the speed, and may cause instability at the low speeds.
The factored form for the equation for lateral stability is
It can be shown in general that the real root corresponding
to the second factor is negative, and that the real parts of
those corresponding to the third factor are negative. Thus
these factors indicate stability. A consideration of the first
factor shows that stability requires that EI and DI be of the
same sign. It is found that DI can be universally regarded as
being positive. Thus EI must be rendered positive. This
condition is the most difficult to obtain in construction.*
120. As an example of the accuracy of the approximate
factorization we shall consider one of the cases considered
by Hunsaker.f The data obtained by experiments are:
i = angle of attack = i.
Velocity = 79 miles an hour, U= 115.5 foot-seconds.
7^ = 55.9 slugs, K B = 34-
X u =.i2&, X w .162, M w 1.74
This gives 4 = 34, =289, C = 8 34 , =115, E = 3 i, BCD-AD*
iSXio 6 . Hence the machine is longitudinally stable.
* For further discussions of the influence of design, and general consideration
of this question, see Bairstow, loc. cit. Hunsaker (i), (2.) Cowley and Evans,
Chap. XII.
t (i). Article 13, Case I.
STABILITY 125
The equation for longitudinal oscillations is
The approximate factorization from Bairstow's formulae is
(X 2 +8.5X+2 4 .5)(X 2 +.i2 5 X+.o374)=o.
The roots obtained from the first factor are
27T
which give the short oscillation of period -- = 2.5 seconds.
2.54
The roots obtained from the second factor are
X= -.063 .183*,
which give the long oscillations of period 34.3 seconds.
More accurate values of the roots of the quartic for the
same machine are given by Wilson,* and are
X= 4.180 2.4302',
X= .0654 .1872.
121. Effect of Gusts. We have so far considered the dis-
turbance occurring while the machine is in still air. As the
atmosphere is in continual and irregular movement the effect
of gusts must be considered. A rather comprehensive study of
this nature has been made by Wilson.* The actual nature of
gusts that occur in the air can obviously not be represented by
mathematical means, because of our lack of knowledge of the
exact conditions and fluctuations in the atmosphere. How-
ever, we can assume certain " mathematical gusts " which
would seem to have as unstabilizing an effect on an airplane
as an actual gust, and by becoming assured of a satisfactory
behavior of the machine in such mathematical gusts of widely
different nature, we can regard the machine as a vehicle possess-
ing an air-worthiness sufficient for the conditions which it may
* E. B. Wilson, "Theory of an Aeroplane Encountering Gusts." First Annual
Report of the (American) Advisory Committee on Aeronautics, Report No. i,
part 2, p. 58. The quartic given by Wilson is
34\ 4 +288.7X 3 +833.oX 2 +ii5.iX+3i.i8 = o.
126 THE DYNAMICS OF THE AIRPLANE
in practice be expected to encounter. The equations that have
been developed in what precedes give the basis for an investiga-
tion of the nature mentioned.
We shall merely indicate the method, without going into
the details of the development, for considerable calculation is
involved. Reference can be made to the detailed and clear
report by Wilson.
122. Suppose first that the machine is flying horizontally
and encounters a gust moving parallel to the x-axis. We think
of it as being caused by a motion of the air parallel to the -axis,
with veolocity i, which we shall take to be positive when
along the negative #-axis. A displacement of the machine
results, and the gust is no longer directly along the rr-axis.
But assuming that all displacements are relatively small, we
shall say that the gust continues unchanged along the #-axis.
The machine has an altered velocity along the s-axis itself, of
amount u, according to our former notation. Therefore the
total change in relative wind is u-\-u\. This is consequently
the quantity by which we should multiply the resistance
derivative X u in the equations of motion.
In the general case we shall assume that the gust has com-
ponents ui, vij wi } pij qij r\. The equations for the longi-
tudinal motion will then be obtained by altering, as indicated
above, the quantities by which the various resistance deriv-
atives are multiplied. We have then for the equations of motion:
-MuU-M w w+(k B 2 D 2 -M q D)d = M u ui +M w u>i +M q qi.
In these equations, u\, vi, . . . . r\ are supposed to be known
functions of the time which vanish for the moment when the
gust commences. The solution will consist of the complimen-
tary function and the particular integrals. The first step is to
solve the system of equations algebraically. We shall take
the result for u as typical. We find
Aw
STABILITY 127
where A is the determinant used before, and AI, A2, AS, are
determinants obtained from A by replacing the elements of the
first column by certain of the resistance derivatives. These
determinants contain the operator Z>; they are to be expanded
as though D were an algebraic quantity, and then applied as
operators to the known functions i, w\, q\. The form of the
determinants makes a general development very difficult.
For a particular machine the knowledge of the resistance
derivatives allow the operators AI, A2, AS to be completely
determined. We can then write as a final equation,
where u(t) is a known function of t as soon as we have assumed
a character for the gust. Similar equations exist for w and 0.
Represent the particular integrals by 7 W , I w , I 9 , respectively.
We then have for the final solutions of the equations:
where Xi, \ 2 , Xa, X4 are the roots of the quartic that gives the
longitudinal motion, and the quantities Cy are constants. These
constants must be determined so that u, w, 6 all vanish for
/ = o, the time at which the gust commences. The fact that
the expressions obtained on dropping /, I W} I 0) must satisfy
the differential equations enables us in the first place to deter-
mine the ratios en : c 2i : c 3 i (i=i, 2, 3, 4). The solutions
can then be expressed in terms of en, Ci 2 , c^, c^. The
values of these are to be determined so that u, w, 0, q = dB/dt
are all zero for t = o. This gives us four equations involving
the constants and the four quantites 7o, 7^, 7 e0 , I'M, which
are the values of the particular integrals for / = o. While
imaginary quantities occur in the process of the work, the
final result can be put in a trigonometric form free from imag-
inaries. The details of this development are given in Wilson's
report, articles 2, 3, 4.
128 THE DYNAMICS OF THE AIRPLANE
123. As an illustration of the application of the method
we shall take the first gust treated by Wilson, for the machine
referred to in 120. Let the gust behead-on, and represented
by
Ui=J(i-e - 2< ), wi=qi=o.
It is seen that the gust increases slowly from o to / in intensity.
The values of the particular integrals found by Wilson are:
- 2 <), /. /o=-. 753-^
I M = -.c82/,
I = - .00495/0 ~- 2 ', /flo = - .0049/,
// = . 00099/0 ~- 2 ', I<' = -00099/.
The complete solution of the differential equations then gives:
w = /0-- 065 %622 cos .i87/4-.63o sin .1870
; = /0~ 4 - 18 '( .004 cos 2.43/4- .003 sin 2.43*)
-/0-- 065 %o78 cos .187/4- .059 sin .187*) + .o82/0-- 2 ',
= /0~- 0654 '(.oo495 cos .i87/ .0031 sin .iSy/) .00495/0" ' 2t -
The effect of the gust is given by Wilson as follows: " (i) The
machine takes up an easy slowly damped oscillation in u of
amplitude about 89 per cent of /; after the oscillation dies
out the machine is making a speed / less relative to the ground
and hence the original relative speed to the wind. (2) There is a
rapidly damped oscillation in w of rather small magnitude and
a slowly damped one of about 10 per cent of /, the final con-
dition being that of horizontal flight. (3) There is a slow
oscillation in pitch of about .oo58/ radians, or about -32/ .
If the magnitude of / is great, the pitching becomes so marked
that the approximate method of solution can no longer be
considered valid a gust of 20 foot-seconds causing a pitch
of some 6. As the period is long (about one-half minute),
the pilot should have ample time to correct the trouble before
it produces serious consequences."
STABILITY 129
124. During the interval that elapses between the com-
mencement of the gust and the acquirement of the subsequent
steady motion the altitude of the machine changes. It is
possible to determine this change. Let the vertical velocity
be represented by -~, being measured upwards. Then, by
dt
resolving u and w along the vertical we have
~ = w cos 6+(V+u) sin 9.
dt
Hence, approximately, the change in altitude is
$= ( (w+Vti)dt.
Jo
For the machine and gust considered this becomes
' 65 %5 cos .I87/-.4 sin .
= J f
Jo
sin .i8yO + 2. 5^-^-3. 5].
When the steady motion has been reached the limit of this is
3.57. For a head-on gust / is negative. Hence the machine
would rise 70 feet on encountering a head-on gust of magnitude
20 foot-seconds.
APPENDIX
1. In the fundamental equation,
F = kAV 2 ,
that we have used for the air pressure, the value of the constant
k depends upon the units we are using. Thus for pressure on
a flat plane, we have:
F = .oo$oAV 2 , if A is in sq. ft., V in mi. per hr., F in Ibs.
F = . 00143^4 F 2 , if A is in sq. ft., V in ft. per sec., F in Ibs.
F=. 075^4 F 2 , if A is in sq. m., V in m. per sec., F in kg.
F = .0057^4 F 2 , if A is in sq. m., F in km. per hr., F in kg.
In case we wish to compare the results that different
experimenters obtain for certain constants, it is necessary to
properly take account of the units employed, before an opinion
can be formed as to the agreement of the results. Similarly,
if we wish to adapt to English units the lift and drag coefficients
for a wing which has been tested in a wind tunnel where metric
units are employed, it is necessary to have a ready means of
making the required transformations.
2. Let us write
where p denotes the density of the air. The dimensions of p,
A, V 2 are respectively, ML~ 3 , L 2 , L 2 T~ 2 . Therefore the dimen-
sions of p^4F 2 are MLT~ 2 . But the dimensions of force, F }
are also MLT~ 2 . Consequently the constant C is dimensionless,
that is, it is an absolute constant, independent of the units
131
132 APPENDIX
employed, provided the system of units is self consistent, that
is, the unit of force is the force required to give to a unit of
mass a unit velocity in a unit of time. Therefore through the
use of the air density we can compare the results of experiments
conducted in different units, or can readily adapt to one system
a series of measurements made in another system of units.
We need merely write the constant k (K y or K*), used before,
in the form,
k=Cp,
where p is the density of the air in the units employed.
3. We shall take some examples.
(1) Take the meter as unit of length, the second as unit
of time, and the weight of a kilogram as the unit of force. In
order to have consistent units, the unit of mass is then 9.8
kilograms. The density in C.G.S units of dry air at 15.6 C.
(60 F.), and pressure of 76 cm. of mercury is .001225. Hence
the mass of a cubic meter is .001225 Xio 6 /io 3 = 1.255 kilo-
grams. In the system we have adopted we have therefore
for the density p = 1.225/9.8 = .125. Therefore,
or C = Sk.
In i we had k = .075 for the units employed here. Hence
C = .6oo.
(2) Take the foot as a unit of length, the second as unit
of time, and the pound as unit of force. The unit of mass is
32.2 Ibs. (a slug). The density of the air is now .00238. Hence
& = .002386*, or C = 42o.2&.
In i we had = .00143 for the units used here. Hence
C = .6oi, which agrees with the results given above for metric
units.
(3) Take the same units for force and length as in (i),
but take the kilometer per hour as the unit of velocity. (This
is not a consistent system of units.)
We write
APPENDIX 133
But
F = .i25C4(Fm./sec.) 2
= . 1 2$CA (1000/3600 X V km./hr.) 2
= . 009604 (Fkm./hr.) 2 .
Hence
' = .00960 or 0=104.2*'.
If we take C = .6oo we have '=.0057, which is the coeffi-
cient given in i for the units in question.
(4) Take the same units as in (2), except that we express
velocities in miles per hour.
We write
F = '.4 (F mi./hr.) 2 .
But
F = . 0023804 (Fft./sec.) 2
= . 0023804 (5280/3600 X F mi./hr.) 2
= .oo5i204(F mi./hr.) 2 .
Hence
*' = . 005120, or 0=195.3*'.
Taking = .6oo we find ' = .0030, which agrees with the
value given in i.
As an illustration of the use of the results obtained we take
the following problem. The lift coefficient on a certain wing
at an angle of attack of 4 is .001455, the units being the pound,
the square foot, and mi./hr. What is the value of the coefficient
if the units are the kilogram, the square meter, and the meter
per second?
We have for the value of the absolute constant by (4),
(for this wing at this angle of attack),
0= 195.3 X .001455= .2841.
By (i) the value of the lift coefficient in the new units would
be
.2841/8 = .0355.
(To make a change of the sort considered here we evidently
multiply by 195.3/8 = 24.4. All types of changes of units
134
APPENDIX
likely to occur can be similarly easily obtained from the results
above.)*
II
4. In considering flight at different altitudes it has been
necessary to consider the varying density of the air. For known
conditions at the surface, that is, known surface pressure and
temperature, the pressure, density, and temperature at any
altitude can be calculated. The formulae for this purpose can
be found in texts on physics, or more extended aeronautic
treatises. We shall merely give a table showing the variation
for certain assumed surface conditions. The ratio n(z) is that
of the pressure at altitude z to the pressure at the surface of
the earth, and the quantity p(z) is the ratio of the air density
at altitude z to the surface density. It will be seen that for
moderate altitudes these two quantities are approximately
equal.
A l4-lf 11/^A
TVvrviT-v
Pressure.
Density.
Altitude,
Feet.
icinp.,
Fahr.
Inches,
Mercury.
Ratio,
?(*).
Slugs.
Ratio,
p().
48
30.0
i .000
.00246
i .000
2,000
44
28.1
0-937
.00232
940
4,000
4i
26.2
873
.00215
-875
6,000
39
24.4
813
.00201
.815
8,000
36
22.6
753
.00187
.760
10,000
30
2O.9
.696
.00175
.710
12,000
25
JQ-3
645
.00163
.665
14,000
20
17.9
.596
.00151
635
16,000
12
16.5
550
.00144
.600
18,000
3
iS-2
.508
.00133
540
20,000
-5
14.0
.467
.00125
510
22,000
14
13.0
* -432
.00118
.480
24,000
-23
12.
.400
.00108
.440
* For a discussion of the question of units see Everett's "Illustrations of the
C. G. S. System of Units with Tables of Physical Constants," Macmillan & Co.,
London and New York, 4th edition, 1891.
APPENDIX
135
III
5. The following table is given for the purpose cf changing
velocities from miles per hour to feet per second, and con-
versely.
mi./hr.
ft./sec.
ft./sec.
mi./hr.
30
44.00
So
34-09
40
58.66
60
40.90
5
73-32
70
47-72
60
88.00
80
54-54
70
102.66
90
61.37
80
117.32
IOO
68.18
90
132.00
no
74-99
IOO
146.66
1 20
81.81
no
161.36
130
88.62
140
95-44
! I5
102. 26
IV
6. References. The literature on the subject of Aviation
has grown with great rapidity in the past few years. Although
no attempt at completeness is made in the following list of
references it is hoped that the different aspects of the subject
are adequately covered. The title of a work indicates whether
it is of general or special character.
1. Cowley and Evans, Aeronautics in Theory and Experiment,
Longmans, Green & Co., New York, 1918.
2. H. Shaw, A Textbook on Aeronautics. J. B. Lippincott Co.,
Philadelphia, 1919.
3. F. W. Lancaster, Aerodynamics.
4. G. Eiffel, La Resistance de 1'Air, H. Dunod et E. Pinat,
Editeurs, Paris, 1911.
5. M. L. Legrand, La Resistance de 1'Air, Librairie Aero-
nautique, 40 rue de Seine, Paris.
6. A. See, Les Lois experimentales de 1'Aviation, Librairie
Aeronautique, Paris.
136 APPENDIX
7. G. Greenhill, The Dynamics of Mechanical Flight, D. Van
Nostrand Company, New York, 1912.
8. A. W. Judge, The Properties of Aerofoils and Aerodynamic
Bodies, James Selwyn & Co., London, 1917.
9. E. B. Wilson, Aeronautics, John Wiley & Sons, Inc., 1920.
10. A Klemin and T. H. Huff, Course in Aerodynamics and
Aeroplane Design, in Aviation and Aeronautical Engineer-
ing, Gardner Moffat Co., New York, commencing in
Vol. i, No. 2, Aug. i, 1916; also published separately.
11. G. H. Bryan, Stability in Aviation, The Macmillan Co.,
London, 1911.
12. R. Devillers, La Dynamique de P Avion, Librairie Aero-
nautique, Paris, 1918.
13. G. De Bothezat, Etude de la Stabilite de 1'Aeroplane,
H. Dunod et E. Pinat, Paris, 1911.
14. R. Soreau, L'Helice Propulsive, Librairie Aeronautique,
Paris, 1911.
15. Duchene, The Mechanics of the Aeroplane, translated by
J. H. Ledeboer and T. Hubbard, Longmans, Green &
Co., 1916.
1 6. Smithsonian Miscellaneous Collection, Vol. 62, 1916, Wash-
ington, D. C.
17. Reports of American Advisory Committee on Aeronautics.
1 8. Reports of British Advisory Committee on Aeronautics.
INDEX.
(The numbers refer to pages)
Aerofoil, 7.
Ailerons, 56, 104.
Altitude, effect of, 21, 77.
Angle, of attack, 5.
economical, 26.
optimum, 22.
for maximum power in ascent,
45-
for maximum traction in ascent,
45-
for minimum vertical velocity
in ascent, 47.
for minimum vertical velocity
in descent, 40.
Aspect ratio, 6.
Asymmetric oscillations, 117.
Bairstow, 121, 122, 123.
Banking, angle of, 53.
Body resistance, 15.
Bryan, 108.
Camber, 7.
Cambered wing, 7.
Ceiling, 49, 86.
determination of, 50, 88.
Center of pressure, 6, 12.
behavior of, 6, 13.
Characteristic curves for wing, 8.
Chord, 7.
Circular descent, 59.
Descent, 36.
rectilinear, 33.
Detrimental surface, 15.
Drag, 6.
Duchemin, formula of, 5.
Economical angle, 26.
Efficiency of propeller, 77.
Elevator, 100.
Equations of
ascent, 43.
circular descent, 59.
horizontal flight, 20,
stability, 114.
turning, 53.
Euler, 108.
Fineness, 23.
Fin, 104.
Gap, 14.
Gusts, effect of, 125.
Helical descent, 59.
Hunsaker, 123, 124.
Inclination, in turning, 54.
Inherent stability, 95.
Landing speed, 20.
Leading edge, 8.
Lift, 6.
Loading, 20.
Metacenter, 98.
Metacentric curve, 98.
Model, of wing, 3.
of complete machine, 15.
Motor diagram, 81.
Moving axes, 104.
Newton, formula of, 5.
Nose, 8.
137
138
INDEX
Optimum angle, 22.
Oscillation, asymmetric, 117.
symmetric, 117.
Pitch of propeller, 71.
Pitching, 97.
Polar diagram, 10.
Power, absorbed by propeller, 82.
total, 29.
used in ascent, 46.
useful, 25.
Pressure, on cambered wing, 7, n.
on plane, 4, 5.
Radius of action, 90.
Resistance derivatives, 116.
Resistance of body, 15.
Rolling, 97.
Rotary derivatives, 116.
Routh, 108, 109, 106.
Rudder, 56, 104.
Stability, dynamic, 97.
lateral, 103.
longitudinal, 97.
in rolling, 103.
static, 96.
Static stability, 96.
Stagger, 14,
Steady motion, 115.
Supercharge, 89.
Symmetric oscillations, 117.
Spiral instability, 106.
Tail plane, 99.
Thrust of propeller, 75.
Time of ascent, 50.
Total power, 29.
Traction, 22, 33, 44.
Turning, 52.
Useful power, 25.
Velocity in
ascent, 44, 47.
circular descent, 44, 47,
descent, 39.
horizontal flight, 20,
turning, 54.
Vertical velocity in
ascent, 44, 49.
descent, 39.
}
Wilson, 125.
Wind tunnel, 3.
Yawing, 97.
^ ^
Wiley Special Subject Catalogues
For convenience a list of the Wiley Special Subject
Catalogues, envelope size, has been printed. These
are arranged in groups each catalogue having a key
symbol. (See special Subject List Below). To
obtain any of these catalogues, send a postal using
the key symbols of the Catalogues desired.
.
I Agriculture. Animal Husbandry. Dairying. Industrial
Canning and Preserving.
2 Architecture. Building. Concrete and Masonry.
3 Business Administration and Management. Law.
Industrial Processes: Canning and Preserving; Oil and Gas
Production; Paint; Printing; Sugar Manufacture; Textile.
CHEMISTRY
' 4a General; Analytical, Qualitative and Quantitative; Inorganic;
Organic.
4b Electro- and Physical; Food and Water; Industrial; Medical
and Pharmaceutical; Sugar.
aOI
CIVIL ENGINEERING
5a Unclassified and Structural Engineering.
5b Materials and Mechanics of Construction, including; Cement
and Concrete; Excavation and Earthwork; Foundations;
Masonry.
5c Railroads; Surveying.
5d Dams; Hydraulic Engineering; Pumping and Hydraulics; Irri-
gation Engineering; River and Harbor Engineering; Water
Supply.
CIVIL ENGINEERING Continued
5e Highways; Municipal Engineering; Sanitary Engineering;
Water Supply. Forestry. Horticulture, Botany and
Landscape Gardening.
6 Design. Decoration. Drawing: General; Descriptive
Geometry; Kinematics; Mechanical.
ELECTRICAL ENGINEERING PHYSICS
7 General and Unclassified; Batteries; Central Station Practice;
Distribution and Transmission; Dynamo-Electro Machinery;
Electro-Chemistry and Metallurgy; Measuring Instruments
and Miscellaneous Apparatus.
'emi
niv'X93'x < I i >iO
8 Astronomy. Meteorology. Explosives. Marine and
Naval Engineering. Military. Miscellaneous Books.
MATHEMATICS
9 General; Algebra; Analytic and Plane Geometry; Calculus;
Trigonometry; Vector Analysis.
MECHANICAL ENGINEERING
lOa General and Unclassified; Foundry Practice; Shop Practice.
lOb Gas Power and Internal Combustion Engines; Heating and
Ventilation; Refrigeration.
lOc Machine Design and Mechanism; Power Transmission; Steam
Power and Power Plants; Thermodynamics and Heat Power.
11 Mechanics.
12 Medicine. Pharmacy. Medical and Pharmaceutical Chem-
istry. Sanitary Science and Engineering. Bacteriology and
Biology.
MINING ENGINEERING
13 General; Assaying; Excavation, Earthwork, Tunneling, Etc.;
Explosives; Geology; Metallurgy; Mineralogy; Prospecting^:
Ventilation.
14 Food and Water. Sanitation. Landscape Gardening.
Design and Decoration. Housing, House Painting.
RETURN ENGINEERING LIBRARY
TO ^ 642-3366
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2
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ALL BOOKS MAY BE RECALLED AFTER 7 DAYS
Overdues subject to replacement charges
DUE AS STAMPED BELOW
UNIVERSITY OF CALIFORNIA, BERKELEY
FORM NO. DD1 1 , 80m, 8/80 BERKELEY, CA 94720
VC 33686'
UNIVERSITY OF CALIFORNIA LIBRARY