Fig. 17 shows a curve (Maxim) giving the lift and drift
FIG. 16. The aerocurve. Angle of attack and entry, trailing angle.
in Ibs. per sq. ft. for an aeroplane shaped as shown in
Fig. 18.
In view of the fact that the curves of pressure dis-
tribution for aerocurves are different from those for aero-
AEROCURVES.
planes, we should expect the travel of the centre of pressure
to be different.
It is found with an aerocurve that the centre of
II
r
9
jQ
i:
rn length a^ecT.
FIG. 17. The aerocurve. Curve showing the lift and drift. (Maxim.)
pressure travels up towards the front edge with decrease
of inclination, until a certain critical angle is reached, after
which the centre of pressure travels backward with a
further decrease of inclination.
In Fig. 19 is plotted the locus of the centre of pressure
of an aerocurve, having a aspect ratio of 1^ and a curvature
FIG. 18. Aerocurve used in Fig. 17-
of yVth the span. Curve 1 was obtained when the hollow
was downwards and curve 2 when the hollow was upwards.
In curve 1 the reversal of the centre of pressure is clearly
18
ELEMENTARY AERONAUTICS.
shown, while in curve 2 it is seen that the centre of pres-
sure advances continually until it is at last in front of the
Distance. frm Front* edge. *
FIG. 19. Centre of pressure curve for an aerocurve with the hollows
downwards and upwards.
80*
7*
60*
SO'
40
30*
to
10
/
/
/
f
/
^
/
^~"
. -
^ - *
~*~~-
{-
>
:
Z. - K -3
Distance C of Pre* -from
Leoo/ir?^ Cc4e.
FIG. 20. Centre of pressure for ship-shaped sections
and aerocurves.
leading edge of the plane. This great advance is largely
due to the effect of the head resistance.
Fig. 20 shows a curve obtained by Prof. Rateau for a ship-
AEROCURVES.
19
shaped section, which is very similar to the curve obtained
for a certain shape of aerocurve.
The travel of the centre of pressure, of course, varies
FIG. 21. The flow of air about an aerocurve.
with the shape of the aerocurve, and until the matter is
thoroughly investigated in the laboratory, it is useless to
attempt to give a formula connecting the travel with
curvature and inclination.
The reversal of travel of the centre of pressure greatly
affects the design of flying machines, because it makes it
impossible to obtain automatic stability with an arc-shaped
aerocurve placed with the hollow downwards. This
reversal is due to the fact that at small angles the wind
strikes the upper side instead of the lower side of the aero-
curve, as shown in Fig. 21, and thus, the front portion
FIG. 22. The centre of pressure on aerocurves. (Wrights.)
which is the most effective part in the case of a flat plane,
altogether ceases to lift.
The Wright Brothers, in one of their early gliders
20 ELEMENTARY AERONAUTICS.
(1901), having given the plane a curvature of T2 -,
that the machine was very difficult to control. They were
in doubt as to the reason for this until, when flying the
machine as a kite, they noticed that in light winds it flew
as shown in the upper figure (Fig. 22), but that, as the
wind became stronger and the angle of incidence less, the
machine flew as shown in the middle figure, with a slight
horizontal pull. When the wind became much stronger,
it took the position shown in the lower figure, with a
strong downward pull.
Thus, it was evident that, in the first case, the centre
of pressure was in front of the centre of gravity ; in the
second the centres coincided ; while in the third, the centre
of pressure was behind the centre of gravity. The curva-
ture was then reduced, and complete success obtained.
STREAM LINES
The explanation of the superior lift of an aerocurve
over an aeroplane, is due to the fact, as we have seen, that
it is a stream line surface.
A stream line is the locus of the successive positions of
a particle of moving fluid, and it must always be a con-
tinuous curve, since it is impossible to make a fluid instantly
change its direction of flow. If the body is so shaped that
it has sharp corners or recesses, the fluid flows past these,
leaving pockets, and forming what are known as surfaces
of discontinuity.
Now, the total energy of a pound of fluid = Potential
energy + Pressure energy + Kinetic energy
. . whenever we have a change in the velocity of a fluid, it
follows that this can only be derived at the expense of the
pressure, or the potential energy.
The pressure and the potential energy are generally
not very great (in the cases we are considering, the
potential energy cannot be utilised, and may therefore be
neglected), and, therefore, the force available for changing
the direction of flow is not great.
Therefore, since force = mass times acceleration, the
change of the direction of flow cannot be made great.
Hence, the reason for the formation of surfaces of dis-
continuity.
AEROCURVES.
21
Fig. 23 is a diagram illustrating the formation of sur-
faces of discontinuity about a rectangular bar.
The air flows in the direction of the arrow until it
meets the bar. It then divides on the front edge. As the
air is unable to turn sharply round the front corners,
it forms a surface of discontinuity, producing a rarefaction
at these places. The air then flows along the sides until
the rear corners are reached, when again a surface of dis-
continuity is formed, producing a rarefaction on the back.
The effect of a surface of discontinuity upon the
pressure diagram is well shown in Fig. 10.
Figs. 24, 25, 26 and 27 show stream line photographs,
taken, with the assistance of Mr A. G. Field, at East
FIG. 23. Stream lines about a rectangular bar.
London College, of the flow of air about variously-shaped
bars.
Fig. 24 shows a rectangle " broadside on."
The air divides at the middle of the front edge, and
forms a surface of discontinuity, with consequent rare-
faction on the back edge.
In Fig. 25 is shown the same rectangle inclined.
The circular section shown in Fig. 26 causes less dis-
turbance to the air than the previous sections, and
therefore offers less resistance to motion. Again, there is
a surface of discontinuity at the back, but this is not so
hard and well-defined as in the previous case.
Fig. 27 shows the flow of air about a triangular bar
with the front edge to the wind. There is a well- defined
surface of discontinuity at the back.
Fig. 28 illustrates the flow of lines of smoke about an
aerocurve inclined at a small angle.
The air divides at the front edge and hugs both sides
as it passes along; its resistance to change of motion
causing a compression on the lower side of the plane and
22
ELEMENTARY AERONAUTICS.
FIG. 24. Stream line flow about a rectangular bar,
broadside on.
FIG. 25. Stream line flow about an inclined
rectangular bar.
AEROCURVES. 23
a, rarefaction or suction on the upper side. As the inclina-
tion is increased, a critical angle appears to be reached,
FIG. 26. Stream line How about a circular bar.
FIG. 27. Stream line flow about a triangular bar.
after which the stream line ceases to follow the upper side
and forms a surface of discontinuity with corresponding
24 ELEMENTARY AERONAUTICS.
eddies (Fig. 29). Also the current divides at a point below
the front edge, as shown in Fig 30, which was taken in
FIG. 28. Stream line flow about an aerocurve at a small inclination.
FlG. 29. Stream line flow about an aerocurve at a large inclination.
water by Prof. Hele Shaw. It will follow that there must
be an inclination at which these two forms of flow merge
AEROCURVES.
25
into one another. According to some experiments made by
Prof. Rateau, it would appear that, at the time the one
system of flow is merging into the other, there is in-
FIG. 30. Stream line flow (water) about an aerocurve.
r
'*
*
<
10'
fo*
FIG. 31. Pressure curves for inclined planes. (Rateau.)
stability. Thus, in Figs. 31 and 32, each curve is in two
distinct and disconnected portions. The hump in Mr
Dines' curves for square and other inclined plates is
probably due to the same cause.
26
ELEMENTARY AERONAUTICS.
The flow of air under a plane is shown in Fig. 33, it
being noticed that there is a slight tendency to rise at the
front edge with this large angle.
f
06
07
ot
to'
Section*
FIG. 32. Pressure curves or inclined planes,
ship-shaped sections. (Kateau )
FIG. 33. Stream line flow beneath plane
The air affected by an aeroplane, that is the field of an
aeroplane, is greater than the air lying in its path. Thus,
in Fig. 34, it will be seen that air, which is considerably
AEROCURVES. 27
above the front edge of the plane, is within the range of
the plane and is deflected downwards. The flow and
FIG. 34. Stream Jine flow above plane.
FIG. 35. Stream line flow between biplanes.
dispersion between the planes of a biplane are shown in
Fig. 35.
28
ELEMENTARY AERONAUTICS.
STREAM LINE SURFACES
A stream line surface is one which does not cause a
surface of discontinuity to be formed.
A fish is one of Nature's stream line surfaces. The
greatest section is in front of the mid-section and the tail
portion has much finer lines than the head portion.
When a body, shaped as shown in Fig. 36, is subjected
to a moving current, it is found that the pressure on the
body at every point is as shown by the direction of the
Stream tme Surface.
FIG. 36.
o-f
Di%contircuitvi
FIG. 37.
FIGS. 36 and 37. Stream line surfaces.
arrows. If the resultant of all these forces is taken, it will
be found to be small and to oppose the direction of motion.
Now, if the back portion is cut away, as shown in Fig.
37, a surface of discontinuity or rarefaction is formed at
the back, and the resultant of all the forces is much greater.
That is, the force required to drive a body so shaped is
much greater than in the case of a stream line body.
Hence it is evident that, in a flying machine, everv
surface should be, if possible, a stream line surface, in order
to avoid loss of energy due to surfaces of discontinuity.
The surface which has been found to require the
minimum power to drive is a long fish-like surface with the
blunt end towards the direction of motion. It was found
OF THE
UNIVERSITY
OF
AEROCURVES. 29
that when the same body was placed with the thin end
opposed to the wind that the driving force was greatly
increased.
RESISTANCE OF BODIES
When we come to consider the resistances of variously
shaped bodies to motion, we find that there is a great lack
of available data.
We may take it as being well-established that, with most
bodies, the resistance varies as the square of the velocity.
The resistance for a normal plane, according to the
Eiffel Tower formula = '003 A V 2
A = area in square feet,
V = velocity in miles per hour.
The following formulae for bars of various sections
have been deduced from experimental data obtained by
Sir Hiram Maxim nearly twenty years ago.
It is not contended that these are exact values, but in
our present state of knowledge they are the best available ;
and, although they will probably be considerably modified,
they should be found useful in the design of machines.
Square bar, with face normal to wind
P = -0039 A V 2 , where
A = area in square feet of one face
V = velocity miles per hour.
Modulus K = (in the expression P = K ('003 A V 2 ) )
= 1-3.
Square bar, with one diagonal in line with the wind
P = -0041AV 2 ,
A = area square feet of one face.
Modulus -1-365.
Round bar
P = -0022AV 2 ,
A = area of cross-section through a diameter.
Modulus = 733.
Ellipse-major diameter = twice minor diameter
P = -0013AV 2 ,
A = area of cross-section through the minor diameter.
Modulus = '43.
Pointed body. Length = 6 times thickness
P = '00016 A V a , where
A = area of section through A B in square feet.
Modulus = -0533.
30
ELEMENTARY AERONAUTICS.
Fish-shaped body
Thick end to wind
P = -000195 A V 2 .
Modulus = '065.
Thin edge to wind-
P = "0005 A V2.
Modulus = '167.
The above formulae may be taken to apply to bodies in
which the smaller dimension varies from \" to 6".
If the body is a long one, and the dimensions small,
the resistance appears to be out of proportion to the size.
This is particularly to be noticed in the case of bracing
wires, and is probably due to the fact that the body
oscillates sideways, and so collides with more air than it
would otherwise do. Therefore, in calculating the re-
- 3- *- 6
FIG. 38. Sections of bars. (Maxim.)
sistance of wires, an ample additional allowance should be
made to meet this extra resistance.
In constructing a machine, special precautions should
be taken to minimise this oscillation by connecting the
wires together where they cross.
SHIELDING
An interesting point in calculating the resistance of a
machine is the shielding effect one body has on another
following in its wake.
This question has been partly explored by Dr Stanton.
He mounted two discs of equal size on the same spindle,
and arranged that the distance between them could be
varied. It was found (Fig. 39) that, when the two discs
AEROCURVES.
31
were close together, the total pressure was only very
slightly greater than that of a single disc. As the discs
were separated, the pressure decreased until a minimum
was reached at a distance apart of 1J diameters, being
then less than 75 per cent, of the resistance of a single
plane.
As the discs were moved still further apart, the
pressure increased.
At a distance of 2'15 diameters, the pressure was again
equal to that on a single disc.
l-tt
73-
I
10
\n oliot
FIG. 39. Shielding of circular plates. (Stanton.)
At a distance of 5 diameters, the total pressure was
1*78 times that on a single plate.
It will be seen from the curve that the total pressure is
not double that on a single plate until the plates are
separated by a distance equal to 10 diameters.
Therefore, in calculating the resistance of struts placed
one behind another, it may be assumed safely that no
shielding takes place if they are at a distance apart equal
to 10 times the smallest dimension.
CHAPTER III
AUTOMATIC LONGITUDINAL STABILITY AND MANUAL
AND AUTOMATIC CONTROL
THE greatest obstacle to the solution of the problem of
flight has been the problem of stability and control.
\ *
FIG. 40. Path of gliders, having
various centres of gravity.
/
rlG. 41. r^atn or gliders, havine:
various centres of gravity.
Its solution was
achieved in a
graduated and up-
ward path by Lilien-
thai, Pilcher, Chanute,
and th|e Wright
Brothers.
If we take a flat
i i i ,1
plane in which the
centres of area and of
gravity coincide, and
launch it into the air, it rolls over and over thus (Fig. 40).
If we weight it at the front, so that the centre of
gravity is not more than one-fifth of the width from the
front, then it takes a header downwards, thus (Fig. 41).
If we then gradually reduce the weight, we shall at
last arrive at a point at which the plane will glide in a
perfectly straight, line A B (Fig. 42).
When this is so, we shall find that the centre of gravity
is between *25 and *3 of the width from the front edge.
32
AUTOMATIC LONGITUDINAL STABILITY. 33
In the first case, the centre of pressure is ahead of the
centre of gravity, and introduces a couple, which rotates
the plane.
In the second case, the centre of gravity is so far ahead
that it drags the plane into the vertical at once.
In the third case, the centres of pressure and of gravity
coincide when the plane is at the natural inclination, with
the result that the plane glides in a straight line.
By utilising the variation of the centre of pressure
with the inclination of the plane, it is possible to make a
glider automatically stable.
A glider can be made to perform two kinds of
oscillations,
(1) It can be made to flutter, or oscillate, about an
axis at right angles to the line of flight ; or
(2) It can be made to " loop the loop," or follow any
one of a number of curved paths.
FIG. 42. Path of gliders, having various centres
of gravity.
1. In the first case, when the glider is travelling at the
natural velocity, and at the correct angle, the centre of
pressure and of gravity coincide ; but if the angle is too
small, the centre of pressure advances and introduces a
couple, which tends to restore the glider- to the correct
angle. Now the glider has a certain inertia, and thus,
when the correct angle is reached and no couple is acting,
it continues to rotate to a greater angle. As soon as the
natural angle is exceeded, the centre of lift travels behind
the centre of gravity and introduces another couple,
tending to reduce the inclination. This sets up a short,
or quick, oscillation.
This oscillation is damped out by the resistance offered
by the air to an oscillating plane.
We have thus explained the reason for the short-pitched
oscillation, and we have established one of the necessary
c
34 ELEMENTARY AERONAUTICS.
conditions of equilibrium; namely, that for automatic
longitudinal equilibrium it is necessary that the centre of
gravity must be within the path of travel of the centre of
lift.
2. If the angle of the glider is the correct one for
horizontal flight at the
p^ \ natural velocity, and
^ the speed is too high,
,. then the lift becomes
I ft * \ too great, and this pro-
I "- -L jr^ \ ^ duces an added vertical
| /S" ^HJ velocity CB. Thus, the
^ CIL, ^^ li ne f travel of the
^C machine is along the
FIG. 43. Path of gliders, having various * ine V"
speeds. Now the effect of
this, as will be seen
from Fig. 43, is to decrease the inclination of the plane
relatively to the line of flight from the angle + /3 to j8.
The centre of pressure will therefore travel forwards,
and introduce a couple which tends to increase the
inclination.
As a result of this added inclination, the inclination of
the path of travel will be still further increased, and if the
original speed be sufficiently great, the glider would " loop
the loop."
If the launching speed of the glider is not sufficient, it
will fall until the
weight and lift are
equal, and opposite.
Thus, instead of
travelling from A
to B, it will travel
down the path A C.
The angle of in-
clination, relatively
to the line of flight, FJG< 44> _ Path of gliders, having various
will therefore be= speeds.
a -f /3. As a result
of this increase of relative inclination, the centre of lift
will travel backwards and will introduce a couple,
tending to decrease the. inclination of the glider. This
will cause the glider to fall until the velocity is sufficient
to right it.
AUTOMATIC LONGITUDINAL STABILITY. 35
SUPPLEMENTARY OR RIDER PLANES
The longitudinal stability may be made more rigid by
the introduction of a second plane set at a distance from
the main plane.
The amount or rigidity of the equilibrium depends in
part upon the distance between the two planes.
4 In the Wrights' machine, there is a small plane placed
horizontally in the front, which, under normal conditions,
rides parallel with the wind.
I the inclination of the plane becomes too small, the
path being along the line D C, the front plane A will be
set at a negative angle to the wind, and thus the reaction
will act downwards.
With decrease of inclination, the centre of lift travels
FIG. 45. The effect of disposition of the planes on
the Wright machine.
forwards, and introduces a couple tending to increase the
angle of inclination.
The effect of the rider plane A is, therefore, to limit or
reduce the travel of the centre of lift forwards.
If the inclination of the machine becomes too great,
the rider plane will also commence to lift.
With increase of inclination, the centre of lift travels
backwards. The backward travel will, therefore, be
limited or reduced by the rider plane.
Thus the effect is, that the travel of the centre of lift
and, therefore, the couple resisting displacement, is reduced
by setting the rider plane at a negative angle. A large
displacement is therefore required to give a comparatively
small restoring couple.
Thus the Wright machine is bound to oscillate through
large angles.
Means are provided for flexing the elevating plane in
either direction to control the machine. Nevertheless, the
fact remains that the Wright disposition of the planes
36
ELEMENTARY AERONAUTICS.
gives just the opposite effect to that which is required for
stability, the principle of stability being, that for maximum
stability the travel of the centre of lift to either side of the
centre of gravity must be a maximum for a given increase
or decrease of the angle of inclination.
It follows, from a similar reasoning, that the stability
may be increased by setting the front plane at a greater or
positive angle with the back plane.
It now remains to consider the methods of increas-
ing the travel of the centre of lift, and of damping
out any oscillation which may be set up by a change of
inclination.
If we place a second plane of a certain aspect and area,
either before or behind the main plane, and parallel to it,
the line of flight becomes more nearly straight.
A'
FIG. 46. Diagram illustrating the effects of rider planes
on the automatic stability.
(1st Case.) Small plane behind,
A small plane, A, behind the main plane B.
With such a disposition, the centre of gravity will be
near the main plane.
Now, behind the main plane, there is a certain wash, or
wake, of disturbed air, the lifting properties of which are
not so good as those of undisturbed air.
Thus, as the tail plane rises to A 1 , the diminution
of the lift is greater than it would be in a clear run
of air.
For the same reason, as the tail falls to A 2 , the
lift increases at a greater rate than it would do in
free air.
This is equivalent to increasing the range of travel of
the centre of lift i.e., of increasing the restoring couple.
It follows that the amount of this additional restoring
couple varies as the length I between the planes.
AUTOMATIC LONGITUDINAL STABILITY.
37
DAMPING
If there were no damping, and the machine received a
displacement, it would go on oscillating for ever about the
neutral line of inclination.
The speed with which an oscillation dies out depends
upon the damping effect of the planes. Now, the resistance
of a normal plane is proportional to the square of the
velocity. If the machine is to oscillate about the point X,
then the velocity v l of the plane B, to that of v 11 of A, is as
a is to b.
Let A = area of plane A, and B that of plane B.
Then Ax6 = Bxa = constant,
. . Damping effect oo
A (v 11 2 b oo A 6 3
i.e., the damping couple for the tail increases as the square
~^t^ ^
FIG. 47. Diagram illustrating the effects of rider planes
on the automatic stability.
of b, or, in other words, the greater the distance between
the planes, the greater the damping effect.
(2nd Case). Small plane in front.
There is less shielding of the back plane when the
small plane rises to the position A 1 . Therefore, as before,
the lift of the plane B will vary at a greater rate than if
the plane were acting in free air.
This will be equivalent to an increase in the travel of
the centre of lift and the restoring couple.
We may increase the sensitiveness by arranging
that the lift of A shall decrease at a less rate than
that of B.
This can be done by giving the front plane A a greater
aspect ratio than the plane B, because the normal pressure
38
ELEMENTARY AERONAUTICS.
on a normal plane is practically independent of the aspect ;
but, as the inclination is reduced, the normal pressure per
unit area is greatest on the plane having the greatest
aspect ratio. Therefore, the greater the aspect ratio of
the plane A, the greater the restoring couple. This couple
may be still further increased by extending the front plane,
so that the maximum interference with the back plane
may be obtained.
The opposite reasoning must be applied when the plane
is at the rear, as in that case it is necessary with decrease
of inclination to decrease the pressure on the tail at a