greater rate than on the main plane.
The shape having the maximum variation of lift with
FIG. 48. Diagram illustrating the effects of rider planes
at the automatic stability.
inclination appears to be a triangle with the apex facing
the wind.
Therefore, we see that for maximum stability,
(1) If the small plane is in front, it should have a
large aspect ratio and a long span.
(2) If behind, it should have a comparatively small
aspect ratio and preferably be triangular
with the apex towards the wind.
(3) In both cases above, the small plane should be
set as far as possible from the main plaue, and
the planes should be set at positive angles
with one another.
The damping effect is the same with the front disposi-
tion as with a back disposition, and therefore it is not
necessary to consider again the theory of this.
AUTOMATIC LONGITUDINAL STABILITY.
39
RANGE OF TRAVEL OF THE CENTRE OF PRESSURE ON
BODIES OF DIFFERENT SHAPES
The amount of longitudinal equilibrium depends upon
the rate of change of the centre of pressure with change
of the angle of inclination.
We have already seen that a fiat plane (1), Fig. 49,
may be made longitudinally stable, owing to the fact that
the centre of pressure approaches the forward edge with
decrease of angle.
With a plane having the concave part downwards, as in
(2), the centre of pressure approaches the forward edge
until a certain angle is reached ; a further decrease in the
angle then causes it to travel backwards. It is therefore
(5)
FIG. 49. Sections of planes in relation to centres of
pressure and automatic stability.
impossible to make the centre of pressure travel to both
sides of the centre of gravity, and hence a plane of this
shape cannot be made stable.
When a concavo - convex plane is turned into the
position (3), the centre of pressure approaches the front
edge continuously with decrease of inclination. The same
change also takes place with the plane shaped as in (4).
Thus, these shaped planes may be made stable. Plane (5),
is unstable as in (2). (5) is unstable as in (2).
MANUAL AND AUTOMATIC CONTROL
In all the cases with which we have dealt so far, the
stability has been automatic without any mechanical
movement of the planes.
Now, the front or back plane may be angularly con-
trolled by the operator, and thus the travel of the centre of
pressure may be governed.
40 ELEMENTARY AERONAUTICS.
This system is known as fore-or-aft control, according
to whether the controlling plane or "elevator" is in front
or behind the main plane.
In some cases, controlling planes, which are cross-
connected, are placed both in front and behind the main
plane. This system is known as fore-and-aft control. It
was first used by Sir Hiram Maxim on his machine of
1893-94. It was used with great success by Curtiss on
the machine which won the first aerial Gordon-Bennett
race. Sir Hiram's latest machine is also provided with
fore-and-aft control, as this system gives the maximum
controlling grip of the air for a minimum longitudinal
length.
AUTOMATIC CONTROL
Instead of the operator controlling the planes by hand,
this may be done by automatic mechanism. This is known
as automatic control.
There are two systems of automatic longitudinal
control,
(1) The gyroscopic, and
(2) The aerodynamic.
The first was invented in 1891 by Sir Hiram Maxim,
and the second was also invented in the same year, and by
an Englishman named Moy. These are described in
Specifications, No. 19228/91 (Maxim), and No. 14742/91
(Moy), respectively.
In the first case, the longitudinal stability is maintained
by means of a gyroscope, which resists a change of motion,
and operates a mechanical relay, so as to throw into action
a large power which operates the stabilising planes.
In the second case, a small vane is made to run in the
air and to operate mechanism.
If the air acts above the plane, it is depressed, and,
vice versa, if below, it is raised.
The subject of automatic control will apparently have
a great future, for, to use an engineer's expression, it will
make the flying machine "fool-proof."
CHAPTER IV
AUTOMATIC LATERAL STABILITY
WE have, so far, only considered the longitudinal, or fore-
and-aft, stability of a glider. Longitudinal stability, how-
ever, is not the only necessary condition for successful
flight. It is necessary to provide means for making a
FIG. 50. Diagram illustrating the various methods of obtaining
lateral stability.
glider stable laterally i.e., along a line at right angles to
the line of flight.
There are at least three well-defined means for doing
this,
(1) By the use of a dihedral angle between the
wings.
(2) By the provision of a vertical plane or planes
above the centre of gravity of the machine.
(3) By a suitable disposition of the centre of gravity.
We will take these in order,
(1) If we take any ordinary sheet of paper and let it
fall, it will roll over and over anyhow ; but, if we bend the
paper about the middle, it will fall straight down without
turning over.
Now, the wings of a machine in flight are set at a
certain angle relatively to the line of flight, as shown in
Fig. 50, where A B is the axis of the machine and C D is
the line of flight.
41
42
ELEMENTARY AERONAUTICS.
(1st Case). To simplify matters, let us assume that the
planes are parallel with the axis of the machine, and that
the machine is rotated about this axis. Let the axis of
the machine when in flight be at an angle a, say,
Thus, from Fig. 51, as the plane AC rotates to the
FIG. 51. Diagram illustrating the various methods of
obtaining lateral stability.
position A 1 C, its inclination to the line of flight will
diminish until, when it is vertical at A' J C, the inclination
will become zero.
On the other hand, the plane C B will increase its
inclination as it rotates to the positions C B 1 , C B 2 , and
will be a maximum and equal to a when it is horizontal.
Now, the speed of the two wiugs relatively to the air-
is the same in both cases. The resultant air pressure will
therefore act at the centre of both wings.
Also, the resultant
force varies with the
inclination of the
plane, thus the force
R 1 will diminish, and
the force R 2 will in-
crease as the rotation
continues.
There is, however,
a further increase of
the force R 2 , and a
corresponding decrease of the force R 1 , owing to the fact
that the sum of the vertical components of R 1 and R 2 ,
must always equal W. If they are not equal, the machine
will fall faster until this is so.
Now, the effect of a greater falling speed is the
.
FIG. 52. Diagram illustrating the various
methods of obtaining lateral stability.
AUTOMATIC LATERAL STABILITY.
43
equivalent of an increase in the angle of inclination of the
planes, and of the axis, to the line of flight, thus,
The inclination is = a to horizontal path A B, but it is =
J-/ 5
FIG. 53. Diagram illustrating the various methods of
obtaining lateral stability.
to downward path A 1 B 1 , therefore we see that, owing
to this effect, the respective changes in R 1 and R 2 will be in
the ratio of the relative angles, and not of the actual
angles i.e., R 2 will be increased and R 1 decreased at a
greater ratio than the variation in the actual angle would
appear to warrant.
Then, if we take moments about the point C, we get,
Couple tending to restore the machine to equilibrium =
R 2 x a - R 1 X a. Where -(a) = half length of wing. It
follows that the maxi-
mum stability effect
from the dihedral
angle may be ob-
tained by having a
dihedral angle of 90,
then, when R 2 is a
maximum, R 1 will be
zero. This is not
advisable, owing to
other reasons.
(2nd Case). We
will now assume that
the wings are in-
clined to the axis, FlG _ 54i _ Diagrammustratingthe variougmethods
and that the axis of obtaining lateral stability.
AB is horizontal.
If we rotate the machine about the axis, it will be seen
that the inclination of the wings always remains the same.
44
ELEMENTARY AERONAUTICS.
Thus, if the path of the machine is kept horizontal, since
the inclination remains the same, the couple introduced by
each wing will be equal and opposite, and there will be no
restoring force.
But, as the machine is rotated, since the horizontal
components of R 1 and R 2 must equal W, it follows that the
inclination of the path of the machine must be increased
i.e., that the machine must fall faster along the line D C
(Fig. 55).
This is, of course, equivalent to setting the axis of the
machine at an angle a with the air. Therefore, when the
axis is rotated, the inclination of the wing on the rising
side to the relative wind is decreased, while that on the
other side is increased.
Thus, as before, it is seen that a couple is introduced,
tending to restore the machine to its normal position.
FIG. 55. Diagram illustrating the various methods of
obtaining lateral stability.
We have taken for the purpose of our argument the
two extreme cases. It follows that a similar reasoning
applies to every intermediate case.
It should be noted that the great objection to the use
of the dihedral angle for obtaining lateral stability is, that
the wedge-shape formation enables the machine to cut its
way downwards through the air, and thus reduces the lift
efficiency. The ideal shape for lift efficiency is undoubtedly
the inverted dihedral angle.
(2) Lateral stability may also be obtained by means of
a suitably disposed vertical plane.
The explanation of this is far more simple than the
case of the dihedral angle.
If the upright plane is at right angles to the main
plane, and above the centre of gravity of the machine,
then, if the axis of rotation is inclined to the line of flight,
as in Fig. 56 (a), it follows that the upright plane will
gradually receive an increasing inclination, which will be
at its maximum when it is horizontal.
AUTOMATIC LATERAL STABILITY.
45
Thus, a normal
reaction R 1 will be
introduced, which
will form a couple
tending to restore
the machine to
equilibrium.
It follows that an
upright plane will
not give lateral
stability if it is so
mounted that, when
the machine is
rotated laterally, it
remains parallel to
the line of flight.
(3) In the third
case, the reaction R
may be resolved
vertically and hori-
zontally.
FIG. 56. Diagram illustrating the various
methods of obtaining lateral stability,
FIG. 57. Diagram illustrating
the various methods of
obtaining lateral stability.
If the centre of gravity is below, then the vertical
component L will equal W, and will form a couple tending
to restore the machine to equilibrium.
46 ELEMENTARY AERONAUTICS.
The horizontal component S will cause the machine to
run sideways. That is to say, the line of flight will be
slightly inclined towards one side of the machine i.e., the
centre of the reaction R wil] travel slightly towards that
side to B, say, and thus still further increase the restoring
couple.
LATERAL CONTROL (MANUAL AND AUTOMATIC)
The lateral stability of a machine may be controlled by
moving small side planes or " ailerons," or by warping the
main plain so that the lower side may be given a greater
inclination and lift. The aileron systems of lateral control
appear to have been first invented by tw 7 o Englishmen,
Boulton and Harte, see 392/68 and 1469/70 respectively,
and the wing-flexing device by the Wright Brothers.
The ailerons may be controlled automatically by means
of pendulums and the like.
Yet another method of controlling a machine laterally
is that proposed by the author, in which a vertical plane,
placed above the centre of gravity of the machine, is
warped or inclined to either side by the aviator.
CHAPTER V
PROPELLERS
SINCE all the power of an engine is supplied through the
propeller, it follows that the efficiency of a machine as
a whole depends upon the efficiency of the propeller.
Thus, great attention should
be paid to the design of a
propeller.
If we take a wedge-shaped
piece of paper, and wrap it
round a cylinder, the upper edge
will form a spiral, or helix (Fig.
58). If a horizontal radius is
kept in contact with the spiral
ABC, it Will Sweep OUt a Spiral FlG . 58 ._ Dia ^ m illustrating
Surface as it rotates around the the principles of the screw
Centre. propeller.
A propeller is
formed by taking a
portion of such a
surface, as shown in
Fig. 59, two blades
being shown repre-
senting two separate
spirals.
When such a pro-
peller screws itself
forward, the air
yields and slips away.
The axial speed of
the propeller is there-
fore not so great as
it would be if there
were no "slip." If
Lx _ - VX_
FIG. 59. Diagram illustrating the principles
of the screw propeller.
no slip occurs, the distance which a screw would move
forward in one revolution is called the pitch, and = AC,
47
48 ELEMENTARY AERONAUTICS.
arid the angle between the two sides is called the pitch
angle.
. ' . Theoretical speed = pitch x revs, per min.
Actual speed = pitch x revs, per min. - slip.
If the pitch of every part of a screw is to remain con-
stant^ the inclination of the surface must increase as the
boss is approached, as is shown in Fig. 60, for two points
A and B.
The theoretical or maximum thrust at any given speed
R (revolutions per minute) is given by the equation
H p = Thrust x speed = Thrust x R x pitch
33,000 ~ 33,000
.'.Theoretical Thmf- H.P.X 33,000
B x pitch.
Fig. 60. Diagram illustrating the
principles of the screw propeller.
If we knew exactly how a propeller acted upon the air,
we should readily be able to calculate its thrust.
In the early days of aeronautics, it was thought that
centrifugal action took place, i.e., it was thought that a
great deal of the air was flung out radially from the
circumference of the propeller, giving no effective reaction
component.
Sir Hiram proved that this view was wrong, by sur-
rounding a propeller with a wire, to which a series of short
ribbons had been attached, and noting the direction of
flow at every point. When his observations were
plotted, the following diagram (Fig. 61) of the lines of
flow was obtained.
PROPELLERS.
49
In the author's
steam line appa-
ratus, ammonium
chloride fumes
were generated
and allowed to
pass into the air
through fine jets,
which were set
at a known dis-
tance apart. The
air was thu s
marked by bands
of smoke, and
from the photos,
which are shown
in Figs. 62 to 66
inclusive, it will
be seen how the
air flows about a
propeller. Fig. 62
FIG. 61. The flow of air about propellers.
FIG. 62. The flow of air about propellers.
50
ELEMENTARY AERONAUTICS.
FIG. 63. The flow of air about propellers.
an end view. This
blade is mounted with
the flat face towards
the left of the photo-
graph (Fig. 66). It,
was found that it
drove the air away
from the flat face in
a cylindrical column,
no matter in which
direction it was
rotated.
The explanation of
this phenomena is
given by Fig. 68. It
will be seen that the
shows a flat-
bladed brass pro-
peller running at
1050 revs, per
min. Fig. 63
shows the radial
infeed on the same
propeller, when
running at 830
revs, per min.
Figs. 64 and 65
show the wooden
propeller which
Sir Hiram Maxim
found to give the
best results. Fig.
66 shows an aero-
dynamical para-
dox which was
discovered by Sir
Hiram Maxim.
The blade is
shaped as shown
in Figs. 67 and
68 ; Fig. 68 being
FIG. 64. The flow of air about propellers.
PROPELLERS.
51
air travels up the front incline and closely adheres while it
travels down the back. Thus the air leaves the plane
with a downward momentum impressed upon it, the
reaction of which is represented by a thrust or a lift.
When the pitch angle of a propeller exceeded 45 at the
circumference, Maxim found that centrifugal or fan-blower
action occurred, the air being driven off in a cone.
We may assume that the air leaves a propeller in a
cylindrical column when the pitch angle is less than 45.
FIG. 65. The flow of air about propellers.
The following deductions may therefore be made :
(1) If the diameter of a propeller = D, and the velocity
of a column of air, when it is running at a
certain speed = V, then the momentum given
to the column of air = MV = thrust, where M =
mass of air driven per second ; the kinetic
energy given to the air or work lost = M V 2 .
Then the ratio of f* , that is, thrust per
ti.r.
horse-power =
Now
MV
i M V 2 V
suppose we reduce the speed of the
52
ELEMENTARY AERONAUTICS.
propeller to one-half. Since only half as much
air is acted upon, thrust = J M x JV.
Work done = J (i M) ( J V) =
FIG. 66. The flow of air about propellers.
MV
thrust
' work done M V 2
~ieT
i.e., just double that of (a).
4
V
(&),
This means that the efficiency of a standing propeller (i.e.,
the ratio of the thrust to the horse-power) varies inversely
PROPELLERS. 53
as the speed. This at first sight appears to be paradoxical,
but it is fully borne out in practice.
(2) Now, let us consider two propellers, having
diameters D and 2D. Let m = mas of unit
FIG. 67. Diagram of an
' aeronautical paradox.
vol. of air. If they both drive a column of
air at the same velocity V, we see,
Thrust of first =M V = DS X V xm) x V.
Thrust of second = (5 (2 D) 2 x V x w) X V.
FIG. 68. The stream line flow about an aerodynamical paradox.
Thrust of second 1
Thrust of first " n D , xV xm
4
54 ELEMENTARY AERONAUTICS.
That is, by doubling the diameter, we get four times the
thrust.
In the same way, by tripling the diameter we shall get
nine times the thrust.
Therefore, the thrust of propellers having the same
pitch, and running at the same speed, varies as the square
of the diameter.
(3) Suppose a constant thrust is desired from pro-
pellers of various diameters, D, 2 D, 3 D.
In the first case,
Thrust = D 2 x V x m) V.
Work done = J M V 2 = J (5 D 2 x V x m) V 2 .
In the second case,
Thrust = (5 (2 D) 2 x V 1 x m) V r
But the thrusts are to be equal,
.-. V 2 = 4V 1 2 .'. V = 2V r
,vV x -JV.
. . work done in the second case =
work done first
work done second
That is to say, the work required to drive a stationary
screw is halved by doubling the diameter. In the same
way it can be shown that, by tripling the diameter, the
work required is divided by three.
Or, in general language,
The work required to drive a standing propeller or
helicopter, thrust being constant, is inversely proportional
to the diameter. Thus, the larger the propeller, the greater
the weight which can be sustained for a given horse-power.
PROPELLERS. 55
In 1809, Sir George Cayley demonstrated that, if it
were possible to construct a screw having a diameter of
200 feet, then a man would be able to support his weight
by means of his own power.
Further, as Sir Hiram has said, if the diameter were
increased to 2000 feet, then the power of a man would
sustain the weight of a horse, and conversely, if the
diameter were reduced to 20 feet, then the power of a
horse would be required to sustain the weight of a man.
The next problem which presents itself is to devise a
formula giving the variation of thrust and horse-power with
the speed and diameter of a propeller.
Let D and 2D = diameters of two screws
N = revs. (in any unit of time).
Now, consider the effect of speed. If we double the speed,
we shall give the air twice the velocity, and we shall also
deal with double the amount, i.e., we shall give it four times
the momentum, and therefore shall get four times the
thrust.
The work done in the first case will be JMV 2 , and in
the second case J(!2M!) (2V) 2 =4MY 2 , i.e., eight times as
much.
If we treble the speed, then, from the same reasoning,
we shall get nine times the thrust, and the work done will
be twenty-seven times as much.
. . the thrust oo N 2
and work done oo N 3 .
Let us now assume that the pitch of both propellers is
the same, i.e., that they both drive a column of air at the
same velocity when running at the same speed.
Since the diameter of the second is double the diameter
of the first, the amount of air driven off by the second is
four times that driven off by the first, because the diameter
of the column of air is doubled, i.e., the thrust of the second
is four times the thrust of the first.
Similarly, if the diameter of the second is three times
the diameter of the first, the thrust will be nine times that
of the first ; therefore, with screws of equal pitch, the
thrust varies as the square of the diameter, and clearly,
the work done varies in the same proportion.
Hence, thrust oo N 2 D 2 ;
work done oo N 8 D 2 .
56
ELEMENTARY AERONAUTICS.
Now, propellers of different diameters, and having the
same pitch, are not similar. Propellers are similar when
the pitch angle is the same.
If the pitch angle is constant, then the pitch varies as
the diameter, i.e., the velocity given to the column of air
varies as the diameter, therefore, if we double the diameter
of the screw, and take equal areas of the column in each
case, since the velocity is doubled, double the amount of air
will be acted upon,
(oitcb
FIG. 69. Diagram illustrating the principles
of the screw propeller.
and each particle
will receive double
the momentum,
therefore, the thrust
will be four times
as great.
Similarly, if the
diameter is trebled,
the thrust will be
nine times as great,
i.e., the thrust per
unit area of the
pitch squared, i.e., to
column is proportional to the
diameter squared.
Gathering our results, we can write :
Thrust oo N 2 D 2 x D 2 oo N 2 D 4
or, introducing a constant K :
Thrust = KN 2 D 4 .... (1).
Now, with regard to the work done, if we take again
equal areas of the column in each case, the diameters being
D and 2D, we get, since the velocity of the column is oo to
pitch, and pitch is oo to diameter :
Work in first case (m X V)V 2 = 1
Work in second case" J(mx 2V)(2V) 2 ~8
When diameters are D and 3D :
Work in first case = 1QV)V 2 _ I_
Work in third case" J(mx 3V)(3V) 2 ~27
i.e., work varies as the cube of the diameter, and
work oo N 3 x D 2 x D 8 x N 8 D 5 ,
or, introducing a constant C :
Power = CN 3 D 6 . . (2).
If, therefore, we determine the thrust and horse- power
for a screw at any one certain speed, we can determine K
PEOPELLERS.
57
and C (the constants in the equation (1) and (2),) and hence,
we can calculate the thrust and horse-power for any speed
of any similar propeller. (By similar, is understood every
propeller having the same pitch angle, and the same relative
shape of blade.) These formulae appear to have been
amply verified by practice, and also by mathematical
investigation.
WALKER AND ALEXANDER'S EXPERIMENTS
Propellers 30 ft. in diameter of the tandem or Mangin
type.
A. 4 blades, 6 ft. wide.
B. 2 blades, 6 ft. wide.
C. 4 blades, 3 ft. wide.
D. 4 blades, 3 ft wide, but 6 ft. radius of blade
removed.
E. 4 blades as A, but at 21 angle.
A to D blades at angle of 12J
E gave greatest thrust per 1 horse-power.
As a result of these experiments, it was found that
(1) Thrust varies as square of revolutions.
(2) Horse-power varies as cube of revolutions.
(3) Thrust per horse-power varies inversely as
revolutions.
RENARD AND KREBES.
Work done
w
Kgm. metres
Revs, per
mins.
Thrust
T.
T3
W" 2
27
17
8
7
70
24
15
69
150
32
26
78
242
35
35
73
364
40
47
78
617
48
64
69
It will be seen that ^p is approximately constant.
The relation -^2= constant, is merely another way of
stating the results obtained by Alexander and Walker.
58 ELEMENTARY AERONAUTICS.
These experiments are a striking justification of the
methods of reasoning previously used.
The beneficial effect of engaging as much air as possible
(because of the relations of the formulae,
Force = MV, and Work lost = JMV 2 ),
is well illustrated by the following examples :
(1) If a revolving stationary propeller is subjected
to a cross current of air, its thrust is increased,
and the greater the speed of the cross current,
the greater the thrust.
(2) When a propeller is first started, its thrust is
greater than when it has been running some
little time, and the air has got into a steady
state of motion. This excess of thrust will
depend entirely upon the speed of acceleration.