THE LIBRARY

OF

THE UNIVERSITY OF CALIFORNIA

LOS ANGELES

FLIGHT WITHOUT FORMULA

THE MECHANICS OF THE AERO- PLANE. A Study of the Principles of Flight. By COMMANDANT DUCHENE. Translated from the French by JOHN H. LEDEBOER, B.A., andT. O'B. HUBBARD. With 98 Illustrations and Diagrams. 8vo. 8s. net.

FLYING : Some Practical Experiences. By GUSTAV HAMEL and CHARLES C. TURNER. With 72 Illustrations. 8vo. I2s. 6d. net.

LONGMANS, GREEN, AND CO.,

LONDON, NEW YORK, BOMBAY.CALCUTTA, MADRAS

FLIGHT WITHOUT FORMULA

SIMPLE DISCUSSIONS ON THE MECHANICS OF THE AEROPLANE

BY

COMMANDANT DUCHENE

OF THE FRENCH GENIE

TRANSLATED FROM THE FRENCH BY

JOHN H. LEDEBOER, B.A.

ASSOCIATE FELLOW, AERONAUTICAL SOCIETY; EDITOR "AERONAUTICS"; JOINT- AUTHOR OF "THE AEROPLANE" TRANSLATOR OF " THE MECHANICS OF THE AEROPLANE"

SECOND EDITION

LONGMANS, GREEN, AND CO.

39 PATERNOSTER ROW, LONDON

FOURTH AVENUE & 30™ STREET, NEW YORK

BOMBAY, CALCUTTA, AND MADRAS

1916

All rights reserved

First Published . . July 1914 Type Reset . . October 1916

TL

S"

JD

I (>

TRANSLATOR'S PREFACE

and equations are necessary evils ; they repre- sent, as it were, the shorthand of the mathematician and the engineer, forming as they do the simplest and most con- venient method of expressing certain relations between facts and phenomena which appear complicated when dressed in everyday garb. Nevertheless, it is to be feared that their very appearance is forbidding and strikes terror to the hearts of many readers not possessed of a mathematical turn of mind. However baseless this prejudice may be — as indeed it is — the fact remains that it exists, and has hi the past deterred many from the study of the principles of the aeroplane, which is playing a part of ever -increasing importance in the life of the community.

The present work forms an attempt to cater for this class of reader. It has throughout been written in the simplest possible language, and contains in its whole extent not a single formula. It treats of every one of the principles of flight and of every one of the problems involved in the mechanics of the aeroplane, and this without demanding from the reader more than the most elementary knowledge of arithmetic. The chapters on stability should prove of particular interest to the pilot and the student, containing as they do several new theories of the highest importance here fully set out for the first time.

In conclusion, I have to thank Lieutenant T. O'B. Hubbard, my collaborator for many years, for his kind and diligent perusal of the proofs and for many helpful suggestions.

J. H. L.

968209

CONTENTS

CHAPTER I

PAGE

FLIGHT IN STILL AIR — SPEED 1

CHAPTER II

PLIGHT IN STILL AIR — POWER 16

CHAPTER III PLIGHT IN STILL AIR — POWER (concluded) .... 35

CHAPTER IV

FLIGHT IN STILL AIR — THE POWER-PLANT . . . . 53

CHAPTER V

FLIGHT IN STILL AIR — THE POWER-PLANT (concluded) . . 70

CHAPTER VI

STABILITY IN STILL AIR — LONGITUDINAL STABILITY . . . 90

CHAPTER VII

STABILITY IN STILL AIR — LONGITUDINAL STABILITY (cotl-

cluded) ..... . . . . 115

CHAPTER VIII

STABILITY IN STILL AIR — LATERAL STABILITY . . .142

CHAPTER IX

STABILITY IN STILL AIR LATERAL STABILITY (concluded)

DIRECTIONAL STABILITY, TURNING . . . . . 161

CHAPTER X

THE EFFECT OF WIND ON AEROPLANES . . . . 183

Flight without Formulae

Simple Discussions on the Mechanics of the Aeroplane

CHAPTER I FLIGHT IN STILL AIR

SPEED

NOWADAYS everyone understands something of the main principles of aeroplane flight. It may be demonstrated in the simplest possible way by plunging the hand in water and trying to move it at some speed horizontally, after first slightly inclining the palm, so as to meet or " attack " the fluid at a small " angle of incidence." It will be noticed at once that, although the hand remains very nearly horizontal, and though it is moved horizon- tally, the water exerts upon it a certain amount of pressure directed nearly vertically upwards and tending to lift the hand.

This, in effect, is the principle underlying the flight of an aeroplane, which consists in drawing through the air wings or planes in a position nearly horizontal, and thus employing, for sustaining the weight of the whole machine, the vertically upward pressure exerted by the air on these wings, a pressure which is caused by the very forward movement of the wings.

Hence, the sustentation and the forward movement of an aeroplane are absolutely interdependent, and the former

1

2 FLIGHT WITHOUT FORMULAS

can only be produced, in still air, by the latter, out of which it arises.

But the entire problem of aeroplane flight is not solved merely by obtaining from the " relative " air current which meets the wings, owing to their forward speed, sufficient lift to sustain the weight of the machine ; an aeroplane, in addition, must always encounter the relative air current in the same attitude, and must neither upset nor be thrown out of its path by even a slight aerial disturbance. In other words, it is essential for an aeroplane to remain in equilibrium — more, in stable equilibrium.

This consideration clearly divides the study of aeroplane flight in calm air into two broad, natural parts :

The study of lift and the study of stability.

These two aspects will be dealt with successively, and will be followed by a consideration of flight in disturbed air.

First we will proceed to examine the lift of an aeroplane in still air.

Following the example of a bird, and in accordance with the results obtained by experiments with models, the wings of an aeroplane are given a span five or six times greater than then1 fore-and-aft dimension, or " chord," while they are also curved, so that their lower surface is concave.* It is desirable to give the wings a large span as compared to the chord, in order to reduce as far as possible the escape or leakage of the air along the sides ; while it has the further advantage of playing an important part in stability. Again, the camber of the wings increases their lift and at the same time reduces their head-resistance or " drag."

The angle of incidence of a wing or plane is the anerle.

* In English this curvature of the wing is generally known as the " camber." On the whole, it would perhaps be more accurate to describe the upper surface as being convex, since highly efficient wings have been designed in which the camber is confined to the upper surface, the lower surface being perfectly flat. — TRANSLATOR.

FLIGHT IN STILL AIR 3

expressed in degrees, made by the chord of the curve in profile with the direction of the aeroplane's flight.

As stated above, the pressure of the air on a wing mov- ing horizontally is nearly vertical, but only nearly. For, though it lifts, a wing at the same time offers a certain amount of resistance — known either as head-resistance or drag * — which may well be described as the price paid for the lift.

As the result of the research work of several scientists, and of M. Eiffel in particular, with scale models, unit figures, or " coefficients," have been determined which enable us to calculate the amount of lift possessed by a given surface and its drag, when moving through the air at certain angles and at certain speeds.

Hereafter the coefficient which serves to calculate the lifting-power of a plane will be simply termed the lift, while that whereon the calculation of its drag is based will be known as the drag.

M. Eiffel has plotted the results of his experiments in diagrams or curves, which give, for each type of wing, the values of the lift and drag corresponding to the various angles of incidence.

The following curves are here reproduced from M. Eiffel's work, and relate to :

A flat plane (fig. 1).

A slightly cambered plane, a type used by Maurice

Farman (fig. 2). A plane of medium camber, adopted by Breguet

(fig. 3).

A deeply cambered plane, used by Bleriot on his No. XI. monoplanes, cross-Channel type (fig. 4).

* The word " drag " is here adopted, in accordance with Mr Archibald Low's suggestion, in preference to the more usual " drift," in order to prevent confusion, and so as to preserve for the latter term its more general, and certainly more appropriate meaning, illustrated in the ex- pression " the drift of an aeroplane from its course in a side-wind," or " drifting before a current." — TRANSLATOR.

4 FLIGHT WITHOUT FORMULA

These diagrams are so simple as to render further explanation superfluous.

.0.00.

0.02 001 000

Drag. Drag.

FIG. 1. — Flat plane. FIG. 2.— Maurice Farman plane.

The calculation of the lifting-power and the head-resist- ance produced by a given type of plane, moving through

FLIGHT IN STILL AIR

the air at a given angle of incidence and at a given speed, is exceedingly simple. To obtain the desired result all that

J0.08

0.00

0.00

0.02 0:01

Drag.

FIG. 3.— Br6guet plane.

O.QO

0 02 0.01 0.00

Drag. FIG. 4.— Bleriot XI. plane.

is needed is to multiply either the lift or the drag co- efficients, corresponding to the particular angle of incidence,

6 FLIGHT WITHOUT FORMULA

by the area of the plane (hi square metres, or, if English measurements are adopted, hi square feet) and by the square of the speed, hi metres per second (or miles per hour).*

EXAMPLE. — A Bleriot monoplane, type No. XI., has an area of 15 sq. m., and flies at 20 m, per second at an angle of incidence of 7°. (1) What weight can its wings lift, and (2) what is the power required to propel the machine ?

Referring to the curve in fig. 4, the lift of this particular type of wing at an angle of 7° is 0-05, and its drag 0-0055.

Hence

T -f. . Square of

Lift. Area. the Speed.

0-05 x 15 x 400

gives the required value of the lifting-power, i.e. 300 kg. Again

Drag. Are, *£™£

0-0055 x 15 x 400 gives the value of the resistance of the wings, i.e. 33 kg.

Let us for the present only consider the question of lift, leaving that of drag on one side.

From the method of calculation shown above we may immediately proceed to draw some highly important de- ductions regarding the speed of an aeroplane. The fore- and-aft equilibrium of an aeroplane, hi fact, as will be shown subsequently, is so adjusted that the aeroplane can only fly at one fixed angle of incidence, so long as the elevator or stabiliser remains untouched. By means of the elevator, however, the angle of incidence can be varied within certain limits.

In the previous example, let the Bleriot monoplane be taken to have been designed to fly at 7°. It has already been shown that this machine, with its area of 15 sq. m. and its speed of 72 km. per hour, will give a lifting-power equal to 300 kg. Now, if this lifting-power be greater than

* Throughout this work the metric system will henceforward be strictly adhered to. — TRANSLATOR.

FLIGHT IN STILL AIR 7

the weight of the machine, the latter will tend to rise ; if the weight be less, it will tend to descend. Perfectly horizontal flight at a speed of 72 km. per hour is only possible if the aeroplane weighs just 300 kg.

In other words, an aeroplane of a given weight and a given plane-area can only fly horizontally at a given angle of incidence at one single speed, which must be that at which the lifting-power it produces is precisely equal to the weight of the aeroplane.

Now it has already been shown that the lifting-power for a given angle of incidence is obtained by multiplying the lift coefficient corresponding to this angle by the plane area and by the square of the speed. This, therefore, must also give us the weight of the aeroplane. It is clear that this is only possible for one definite speed, i.e. when the square of the speed is equal to the weight, divided by the area multiplied by the inverse of the lift. And since the weight of the aeroplane divided by its area gives the load- ing on the planes per sq. m., the following most important and practical rule may be laid down :

The speed (in metres per second) of an aeroplane, flying at a given angle of incidence, is obtained by multiplying the square root of its loading (in kg. per sq. in.} by the square root of the inverse of the lift corresponding to the given angle.

At first sight the rule may appear complicated. Actually it is exceedingly simple when applied.

EXAMPLE. — A Breguet aeroplane, with an area of 30 sq. m. and weighing 600 kg., flies with a lift of 0-04, equivalent (according to the curve in fig. 3) to an angle of incidence of about 4°. What is its speed ?

600 The loading is ——=20 kg. per sq. m.

Square root of the loading— 4 '47.

Inverse of the lift is —=25. 0-04

Square root of inverse of the lift =5.

FLIGHT WITHOUT FORMULA

The speed required, therefore, in metres per second— 4-47 X 5=22- 3 m. per second, or about 80 km. per hour. But if a different angle of incidence, or a different figure for the lift — which is equivalent, and, as will be seen here- after, more usual — be taken, a different speed will be obtained.

Hence each angle of incidence has its own definite speed.

For instance, if we take the Breguet aeroplane already considered, and calculate its speed for a whole series of angles of incidence, we obtain the results shown in Table I. But before examining these results in greater detail, so far as the relation between the angles of incidence, or the lift, and the speed is concerned, a few preliminary observations may be useful.

TABLE I.

			'Speed.
In m.p.s.	In km.p.h.
Lift.	Correspond- ing Angle of Incidence.	! Square Inverse of i Root of Lift. ; Inverse of Lift.
~* >j~o 50	10 >» |1^ .
				J'ilfcr	lit"
				:(? S 3 "8	|*a
1	2	3	4	5	6
0-020	0° (about)	50	7-07	31-6	113-6
0-030	2° ,,	33-3 577	25-8	92-8
0-040	4° ,,	25	5-00	22-3	80-3
0-050	64° „	20	4-47	20-0	72-0
0-060	10° ,,	16-6 4-08	18-2	65-6
0-066	15° ,,	15-2	3-90	17-4	62-6

In the first place, it should be noted that when the Breguet whig has no angle of incidence, when, that is, the wind meets it parallel to the chord, it still has a certain lift. This constitutes one of the interesting properties of

FLIGHT IN STILL AIR 9

a cambered plane. While a flat plane meeting the air edge-on has no lift whatever, as is evident, a cambered plane striking the air in a direction parallel to its chord still retains a certain lifting-power which varies according to the plane section.

Thus, in those conditions a Breguet wing still has a lift of 0-019, and if figs. 4 and 2 are examined it will be seen that at zero incidence the Bleriot No. XI. would similarly have a lift of 0-012, but the Maurice Farman of only 0-006. It follows that a cambered plane exerts no lift whatever only when the wind strikes it slightly on the upper surface. In other words, by virtue of this property, a cambered plane may be regarded as possessing an imaginary chord — if the expression be allowed — inclined at a negative angle (that is, in the direction opposed to the ordinary angle of incidence) to the chord of the profile of the plane viewed in section.

If the necessary experiments were made and the curves on the diagrams were continued to the horizontal axis, it would be found that the angle between this " imaginary chord " and the actual chord is, for the Maurice Farman plane section about 1°, for that of the Bleriot XI. some 2°, and for that of the Breguet 4°.

Let it be noted in passing that in the case of nearly every plane section a variation of 1° in the angle of incidence is roughly equivalent to a variation in lift of 0-005, at any rate for the smaller angles. One may therefore generalise and say that for any ordinary plane section a lift of 0-015 corresponds to an angle of incidence of 3° relatively to the " imaginary chord," a lift of 0-020 to an angle of 4°, a lift of 0-025 to 5°, and so forth.

Turning now to the upper portion of the curves in the diagrams, it will be seen that, beginning with a definite angle of incidence, usually in the neighbourhood of 15°, the lift of a plane no longer increases. The curves relating to the Breguet and the Bleriot cease at 15°, but the Maurice Farman curve clearly shows that for angles of incidence greater than 15° the lift gradually diminishes. Such coarse

10 FLIGHT WITHOUT FORMULA

angles, however, are never used in practice, for a reason shown in the diagrams, which is the excessive increase in the drag when the angle of incidence is greater than 10°. In aviation the angles of incidence that are employed there- fore only vary within narrow limits, the variation certainly not surpassing 10°.

We may now return to the main object for which Table I. was compiled, namely, the variation in the speed of an aeroplane according to the angle of incidence of its planes.

First, it is seen that speed and angle of incidence vary inversely, which is obvious enough when it is remembered that in order to support its own weight, which necessarily remains constant, an aeroplane must fly the faster the smaller the angle at which its planes meet the air.

Secondly, it will be seen that the variation in speed is more pronounced for the smaller angles of incidence ; hence, by utilising a small lift coefficient great speeds can be attained. Thus, for a lift equal to 0-02, at which the Breguet wing would meet the air along its geometrical chord, the speed of the aeroplane, according to Table I., would exceed 113 km. an hour.

If an aeroplane could fly with a lift coefficient of O'Ol, that is, if the planes met the air with their upper surface — the imaginary chord would then have an angle of incidence of no more than 2° — the same method of calculation would give a speed of over 160 km. per hour.

The chief reason which in practice places a limit on the reduction of the lift is, as will be shown subsequently, the rapid increase in the motive -power required to obtain high speeds with small angles of incidence. And further, there is a considerable element of danger in unduly small angles. For instance, if an aeroplane were to fly with a lift of O'Ol — so that the imaginary chord met the air at an angle of only 2° — a slight longitudinal oscillation, only just exceeding this very small angle, would be enough to convert the fierce air current striking the aeroplane moving at an enormous speed from a lifting force into one provoking a fall. It is

FLIGHT IN STILL AIR 11

true that the machine would for an instant preserve its speed owing to inertia, but the least that could happen would be a violent dive, which could only end in disaster if the machine was flying near the ground.

Nevertheless there are certain pilots, to whom the word intrepid may be justly applied, who deny the danger and argue that the disturbing oscillation is the less likely to occur the smaller the angle of incidence, for it is true, as will be seen hereafter, that a small angle of incidence is an important condition of stability. However this may be, there can be no question but that flying at a very small angle of incidence may set up excessive strains in the frame- work, which, in consequence, would have to be given enormous strength . Thus , if it were possible for an aeroplane to fly with a lift coefficient of 0-01, and if, owing to a wind gust or to a manoeuvre corresponding to the sudden " flatten- ing out " practised by birds of prey and by aviators at the conclusion of a dive, the plane suddenly met the air at an angle of incidence at which the lift reaches a maximum — that is, from 0-06 to 0-07 according to the type of plane — the machine would have to support, the speed remaining constant for the time being by reason of inertia, a pressure six or seven times greater than that encountered in normal flight, or than its own weight.

In practice, therefore, various considerations place a limit on the decrease of the angle of incidence, and it would accordingly appear doubtful whether hitherto an aeroplane has flown with a lift coefficient smaller than 0-02.*

It is easy enough to find out the value of the lift co- efficient at which exceptionally high speeds have been attained from a few known particulars relating to the machine in question. The particulars required are :

The velocity of the aeroplane, which must have been carefully timed and corrected for the speed of the wind ;

The total weight of the aeroplane fully loaded ;

The supporting area.

* See footnote on p. 12.

12 FLIGHT WITHOUT FORMULA

The lift may then be found by dividing the loading of the planes by the square of the speed in metres per second.

EXAMPLE. — An aeroplane with a plane area of 12 sq. m. and weighing, fully loaded, 360 kg. has flown at a speed of 130 km. or 36-1 m. per second. What was its lift coefficient ?

O f*(\

The loading= = 30 kg. per sq. m.

12

Square of the speed=1300.

Ofi

Required lift=-?"- =about 0-023.*

Table I. further shows that when the angle of incidence reaches the neighbourhood of 15° (which cannot, as has been seen, be employed in practical flight) the lift reaches its maximum value, and the speed consequently its minimum.

* At the time of writing (August 1913) the speed record, 171 '7 km. per hour or 47'6 m. per second, is held by the Deperdussin monocoque with a 140-h.p. motor, weighing 525 kg. with full load, and with a plane area of about 12 sq. m. (loading, 43'7 kg. per sq. m.). Another machine of the same type, but with a 100-h.p. engine, weighing 470 kg. in all, and with an area of 11 sq. m., has attained a speed of 168 km. per hour or 46'8 m. per second. According to the above method of calculation, the flight in both cases was made with a lift coefficient of about 0'0195. — AUTHOR.

Since the above was written, all speed records were broken during the last Gordon-Bennett race in September 1913. The winner was Prevost, on a 160-h.p. Gnome Deperdussin monoplane, who attained a speed of a fraction under 204 km. per hour ; while Vedrines, on a 160- h.p. Gnome-Ponnier monoplane, achieved close upon 201 km. per hour. The Deperdussin monoplane, with an area of 10 sq. m., weighed, fully loaded, about 680 kg. ; the Ponnier, measuring 8 sq. m., weighed ap- proximately 500 kg. Adopting the same method of calculation, it is easily shown that the lift coefficients worked out at 0'021 and 0'020 respectively. It is just possible that these figures were actually slightly smaller, since it is difficult to determine the weights with any consider- able degree of accuracy. However, the error, if there be any, is only slight, and the result only confirms the author's conclusions. Since that time Emile Vedrines is stated to have attained, during an official trial, a speed of 212 km. per hour, on a still smaller Ponnier monoplane measuring only 7 sq. m. in area and weighing only 450 kg. in flight. This would imply a lift coefficient of 0'0185, a figure which cannot be accepted without reserve. — TRANSLATOR.

FLIGHT IN STILL AIR 13

If the angle surpassed 15° the lift would diminish and the speed again increase.

A given aeroplane, therefore, cannot in fact fly below a certain limit speed, which in the case of the Breguet already considered, for instance, is about 63 km. per hour.

It will be further noticed that in Table I. one of the columns, the second one, contains particulars relating only to the Breguet type of plane. If this column were omitted, the whole table would give the speed variation of any aeroplane with a loading of 20 kg. per sq. m. on its planes, for a variation in the lift coefficient of the planes. It was this that led to the above remark, made in passing, that it was more usual to take the lift coefficient than the angle of incidence ; for the former is independent of the shape of the plane.

The speed variation of an aeroplane for a variation in its lift coefficient can easily be plotted hi a curve, which would have the shape shown in fig. 5, which is based on the figures in Table I.

The previous considerations relate more especially to a study of the speeds at which a given type of aeroplane can fly. In order to compare the speeds at which different types of aeroplanes can fly at the same lift coefficient, we need only return to the basic rule already set forth (p. 7). . It then becomes evident that these speeds are to one another as the square roots of the loading.

The fact that only the loading comes into consideration in calculating the speed of an aeroplane shows that the speed, for a given lift coefficient, of a machine does not depend on the absolute values of its weight and its plane area, but only on the ratio of these latter. The most heavily loaded aeroplanes yet built (those of the French military trials in 1911) were loaded to the extent of 40 kg. per sq. m. of plane area.* The square root of this number being 6-32, an aeroplane of this type, driven by a sufficiently

* The 140-kp. Deperdussin monocoque had a loading of 43-8 kg.

per sq. m.

14

FLIGHT WITHOUT FORMULA

powerful engine to enable it to fly at a lift coefficient of 0-02 (the square root of whose inverse is 7-07), could have attained a speed equal to 6-32 x 7-07, that is, it could have exceeded 44-5 m. per second or 160 km. per hour.

It is therefore evident that there are only two means for increasing the speed of an aeroplane — either to reduce the

30

20

10

00

Lift

0.0/0 0.020 0.030 0,0*0 0.050 0.060 0.076 FIG. 5.

lift coefficient or to increase the loading. Both methods require power ; we shall see further on which of the two is the more economical in this respect.

The former has the disadvantages — contested, it is true —which have already been stated. The latter requires exceptionally strong planes.

In any case, it would appear that, in the present stage of aeroplane construction, the speed of machines will scarcely exceed 150 to 160 km. per hour ; and even so, this result

FLIGHT IN STILL AIR 15

could only have been achieved with the aid of good engines developing from 120 to 130 h.p.* So that we are still far removed from the speeds of 200 and even 300 km. per hour which were prophesied on the morrow of the first advent of the aeroplane, f

In concluding these observations on the speed of aero- planes, attention may be drawn to a rule already laid down in a previous work,! which gives a rapid method of calculat- ing with fair accuracy the speed of an average machine whose weight and plane area are known.

The speed of an average aeroplane, in metres per second, is equal to five times the square root of its loading, in kg. per sq. m.

This rule simply presupposes that the average aeroplane flies with a lift coefficient of 0-04, the inverse of whose square root is 5. The rule, of course, is not absolutely accurate, but has the merit of being easy to remember and to apply.

EXAMPLE. — What is the speed of an aeroplane weighing 900 kg., and having an area of 36 sq. m. ?

900 Loading = — r = 25 kg. per sq. m.

OO

Square root of the loading =5.

Speed required=5x5=25 m. per second or 90 km. per hour.

* In previous footnotes it has already been stated that the Deper- dussin monocoques, a 140-h.p. and a 100-h.p., have already flown at about 170 km. per hour. But these were exceptions, and, on the whole, the author's contention remains perfectly accurate even to-day. — TRANS- LATOR.

f The reference, of course, is only to aeroplanes designed for everyday use, and not to racing machines. — TRANSLATOR.

J The Mechanics of the Aeroplane (Longmans, Green & Co.).

CHAPTER II

FLIGHT IN STILL AIR

POWER

IN the first chapter the speed of the aeroplane was dealt with in its relation to the constructional features of the machine, or its characteristics (i.e. the weight and plane area), and to its angle of incidence. It may seem strange that, in considering the speed of a motor-driven vehicle, no account should have been taken of the one element which usually determines the speed of such vehicles, that is, of the motive-power. But the anomaly is only apparent, and wholly due to the unique nature of the aeroplane, which alone possesses the faculty — denied to terrestrial vehicles which are compelled to crawl along the surface of the earth, or, in other words, to move hi but two dimensions — of being free to move upwards and downwards, in all three dimensions, that is, of space.

The subject of this chapter and the next will be to examine the part played by the motive-power in aeroplane flight, and its effect on the value of the speed.

In all that has gone before it has been assumed that, in order to achieve horizontal flight, an aeroplane must be drawn forward at a speed sufficient to cause the weight of the whole machine to be balanced by the lifting power exerted by the planes. But hitherto we have left out of consideration the means whereby the aeroplane is endowed with the speed essential for the production of the necessary lifting-power, and we purposely omitted, at the time, to

FLIGHT IN STILL AIR 17

deal with the head-resistance or drag, which constitutes, as already stated, the price to be paid for the lift.

This point will now be considered.

Reverting to the concrete case first examined, that of the horizontal flight at an angle of incidence of 7° of a Bleriot monoplane weighing 300 kg. and possessing a wing area of 15 sq. m., it has been seen that the speed of this machine flying at this angle would be 20 m. per second or 72 km. per hour, and that the drag of the wings at the speed mentioned would amount to 33 kg.

Unfortunately, though alone producing lift in an aero- plane, the planes are not the only portions productive of drag, for they have to draw along the fuselage, or inter- plane connections, the landing chassis, the motor, the occupants, etc.

For reasons of simplicity, it may be assumed that all these together exert the same amount of resistance or drag as that offered by an imaginary plate placed at right angles to the wind, so as to be struck full in the face, whose area is termed the detrimental surface of the aeroplane.

M. Eiffel has calculated from experiments with scale models that the detrimental surface of the average single-seater monoplane amounted to between f and 1 sq. m., and that of an average large biplane to about 1| sq. m.* But it is clear that these calculations can only have an approximate value, and that the detrimental surface of an aeroplane must always be an uncertain quantity.

But in any case it is evident that this parasitical effect should be reduced to the lowest possible limits by stream- lining every part offering head-resistance, by diminishing exterior stay wires to the utmost extent compatible with safety, etc. And it will be shown hereafter that these measures become the more important the greater the speed of flight.

The drag or passive resistance can be easily calculated * These figures have since been undoubtedly reduced.

2

18 PLIGHT WITHOUT FORMULA

for a given detrimental surface by multiplying its area in square metres by the coefficient 0-08 (found to be the average from experiments with plates placed normally), and by the square of the speed in metres per second.

Thus, taking once again the Bleriot monoplane, let us suppose it to possess a detrimental surface of 0-8 sq. m. ; its drag at a speed of 72 km. per hour or 20 m. per second will be:

n «? • * Detrimental Square of Coefficient. gurface> ^ gpeed>

0-08 x 0-8 x 400 =26 kg. (about).

As the drag of the planes alone at the above speed amounts to 33 kg., it is necessary to add this figure of 26 kg., in order to find the total resistance, which is therefore equal to 59 kg. The principles of mechanics teach that to overcome a resistance of 59 kg. at a speed of 20 m. per second, power must be exerted whose amount, expressed in horse-power, is found by dividing the product of the resistance (59 kg.) and the speed (20 m. per second) by 75.* We thus obtain 16 h.p. But a motor of 16 h.p. would be insufficient to meet the requirements.

I^or the propelling plant, consisting of motor and pro- peller, designed to overcome the drag or air resistance of the aeroplane, is like every other piece of machinery subject to losses of energy. Its efficiency, therefore, is only a portion of the power actually developed by the motor. The efficiency of the power-plant is the ratio of useful, power — that is, the power capable of being turned to effect after transmission — to the motive power.

Thus, in order to produce the 16 h.p. required for horizontal flight in the above case of the Bleriot mono-

* This is easily understood. The unit of power, or horse-power, is the power required to raise a weight of 75 kg. to a height of 1 in. in 1 second, so that, to raise in this time a weight of 59 kg. to a height of

59 x 20 20 m., we require — =^ — h.p. Exactly the same holds good if, instead

of overcoming the vertical force of gravity, we have to overcome the horizontal resistance of the air.

FLIGHT IN STILL AIR 19

plane, it would be necessary to possess an engine develop- ing 32 h.p. if the efficiency is only 50 per cent., 26-6 h.p. for an efficiency of 60 per cent., etc.

But if the aeroplane were to fly at an angle of incidence other than 7° — which, as already stated, would depend on the position of the elevator — the speed would necessarily be altered. If this primary condition were modified, the immediate result would be a variation in the drag of the planes, in the head-resistance of the aeroplane, in the propeller-thrust, which is equal to the total drag, and lastly, in the useful power required for flight.

Each value of the angle of incidence — and consequently of the speed — therefore has only one corresponding value of the useful power necessary for horizontal flight.

Returning to the Breguet aeroplane weighing 600 kg. with a plane area of 30 sq. m., on which Table I. was based, we may calculate the values of the useful powers required to enable it to fly along a horizontal path for different angles of incidence and for different lifts. The detrimental surface may be assumed, for the sake of simplicity, to be 1-2 sq. m.

The values of the drag corresponding to those of the lift will be obtained from the polar diagram shown in fig. 3.

Table II., p. 20, summarises the results of the calcula- tion required to find the values of the useful powers for horizontal flight at different lift coefficients.

Various and interesting conclusions may be drawn from the figures in columns 8 and 9 of this Table.

In the first place, it will be noticed from the figures in column 8 that the propeller-thrust (equivalent to the drag of the planes added to the head-resistance of the machine, i.e. column 6 and column 7) has a minimum value of 91 kg., corresponding to a lift coefficient of 0-05, and to the angle 6£°. This angle, which, in the case under consideration, is that corresponding to the smallest propeller-thrust, is usually known as the optimum angle of the aeroplane.

20

FLIGHT WITHOUT FORMULA

TABLE II.

	| •« «	Speed Values.	fl	Bfe g^g	§1^' -S g g ^|| g	it-is *-:-£
	&C ^"		3 ^ o 5T1		^	o ~~ '**. ^a
Lift.	II ic s *s		11 n o |	Drag of P (drag x area, 3 x square of	il«^ llfi |sl i1 wsl x	Propeller- T	Useful Power f zontal Flight ( product of figs 3 and 8 divide
m.p.s.	kni.p.h.
1	2	3	4	5	6	7	8	9
0-020	0°	31-6	113-6	0-0022	66kg.	96kg.	162kg.	68
0-030	2°	25-8	92-8	0-0024! 48	64	112	38
0-040	4°	22-3	80-3	0-0032	48	48	96	29
0-050	64*	20-0	72-0	0-0044	53	38	91	24
0-060	10°	18-2	65-6	0-0063	63	32	95	23
0-066	15°	17-4 62-6	0-0118	107	29	136	31

When the lift coefficient is small, the requisite thrust, it will be seen, increases very rapidly, and the same holds good for high lift coefficients.

Secondly, the figures in column 9 show that, together with the thrust, the useful power required for flight reaches a minimum of 23 h.p., corresponding to a lift value of 0-06. The angle of incidence at which this minimum of useful power can be achieved, about 10° in the present case, can be termed the economical angle.

This angle is greater than the optimum angle, which can be explained by the fact that, though the thrust begins to increase again, albeit very slowly, when the angle of incidence is raised above the optimum angle, the speed still continues to decrease to an appreciable extent, and for the time being this decrease in speed affects the useful power more strongly than the increasing thrust ; and the minimum value of the useful power is, consequently, not attained until, as the angle of incidence continues to grow, the

FLIGHT IN STILL AIR 21

increase in the thrust exactly balances the decrease in the

The figures in column 9 again show the great expenditure of power required for flight at a low lift coefficient. Thus, the Breguet aeroplane already referred to, driven by a propelling plant of 50 per cent, efficiency, flying at a lift of 0-05 — that is, at a speed of 72 km. per hour — only requires an engine developing 46 h.p. ; but it would need a 136-h.p. engine to fly with a lift of O02, or at about 113 km. per hour. It is mainly on this account that, as we have already stated, the use of low lift coefficients is strictly limited.

The variations in power corresponding to variations in speed (and in lift) can be plotted in a simple curve.

Fig. 6 is of exceptional importance, for it may be said to determine the character of the machine, and will hereafter be referred to as the essential aeroplane curve.

After these preliminary considerations on the power required for horizontal flight, we may now proceed to examine the precise nature of the effect of the motive- power on the speed, which will lead at the same time to certain conclusions relating to gliding flight *

For this, recourse must be had to one of the most elemen- tary principles of mechanics, known as the composition and decomposition of forces. The principle is one which is almost self-evident, and has, in fact, already been used in these pages, when at the beginning of Chapter I. it was shown that in the air pressure, which is almost vertical, on a plane moving horizontally, a clear distinction must be made between the principal part of this pressure, which is strictly vertical (the lift), and a secondary part, which is strictly horizontal (the drag).

And, conversely, it is evident that for the action of two forces working together at the same time may be substituted

* There is really no excuse for the importation into English of the French term " vol plane," and still less for the horrid anglicism " volplane," since " gliding flight " is a perfect English equivalent of the French. — TRANSLATOR.

22

FLIGHT WITHOUT FORMULA

that of a single force, termed the resultant of these two forces. This proceeding is known as the composition of forces. So, in compounding the vertical reaction con- stituting the lift, and the horizontal reaction which forms

71)

40

3*

id

-

10

Flying Speeds (m/s).

FIG. 6.

The figures on the curve indicate the lift.

the drift, one obtains the total air pressure, which is simply their resultant.

Both the composition and decomposition of forces is accomplished by way of projection. Thus (fig. 7), the force Q,* which is inclined, can be decomposed into two forces,

* A force is represented by a straight line, drawn in the direction in which the force operates, and of a length just proportional to the magnitude of the force.

FLIGHT IN STILL AIR

F and r, vertical and horizontal respectively, by projecting in the horizontal and vertical directions the end point A on two axes starting from the point 0, where the forces are applied. Conversely, these two forces F and r may be

FIG. 7.

FligU-Path. ._

FIG. 8.

compounded into one resultant Q, by drawing the diagonal of the parallelogram or rectangle of which they form two of the sides. We may now return to the problem under consideration.

If we take the aeroplane as a whole, instead of dealing with the planes alone, it will be readily seen that the horizontal component of the air pressure on the whole

24 FLIGHT WITHOUT FORMULA

machine is equal to the drag of the planes added to the passive or head-resistance, the while the vertical component remains practically equal to the bare lift of the planes, since the remaining parts of the structure of an aeroplane exert but slight lift, if at all.* The entire pressure of the air on a complete aeroplane in flight is therefore farther inclined to the perpendicular than that exerted on the planes alone.

If (see fig. 8) the aeroplane is assumed to be represented by a single point O, in horizontal flight, the air pressure Q exerted upon it may be decomposed into two forces, of which the lift F is equal and directly opposite to the weight P, and the drag r, or total resistance to forward movement, which must be exactly balanced by the thrust t of the propeller.

But, supposing the engine be stopped and the propeller consequently to produce no thrust (fig. 9), the aeroplane will assume a descending flight-path such that the planes still retain the single angle of 7°, for instance, which we have assumed, so long as the elevator is not moved, and such that the air pressure Q on the planes becomes absolutely vertical, in order to balance the weight of the machine, instead of remaining inclined as heretofore. This is gliding flight.

Relatively to the direction of flight, the air pressure Q still retains its two components, of which r is simply the resistance of the air opposed to the forward movement of the glider. The second component F is identical to the lifting power in horizontal flight, and its value is obtained by multiplying the lift coefficient corresponding to the angle 7° by the plane area, and by the square of the speed of the aeroplane on its downward flight-path.

Fig. 9 shows that, by the very fact of being inclined, the force F is slightly less than the weight of the machine, but, since the gliding angle of an aeroplane is usually a slight

* For the sake of simplicity, we may consider that the tail plane, which will be hereinafter dealt with, exerts no lift.

FLIGHT IN STILL AIR

25

one, the lifting power F may still be deemed to be equal to the weight of the aeroplane.

Clearly, therefore, every consideration in the first chapter which related to the speed in horizontal flight is equally applicable to gliding flight, so that it may be said that

FIG. 9.

when an aeroplane begins to glide, without changing its angle, the speed remains the same as before.

In fact, horizontal flight is simply a glide in which the angle of the flight-path has been raised by mechanical means.

On comparing figs. 8 and 9 it will be seen that this angle is that which, in fig. 8, is marked by the letter p. If this

26

FLIGHT WITHOUT FORMULAE

angle is represented, as in the case of any gradient, in terms of a decimal fraction, it will be found to depend on the ratio which the forces r and F bear to one another. Hence, the following rule may be stated : —

RULE. — The gliding angle assumed at a given angle of incidence by any aeroplane is equal to the thrust required for its horizontal flight at the same angle, divided by the weight of the machine.

Thus the Bleriot monoplane dealt with in the first instance, which requires for horizontal flight at an angle of 7° a thrust of 59 kg., and weighs 300 kg., would assume on its glide, at the same angle of incidence, a descending

59

flight-path equal to — — , or 0-197, which is equivalent to oOO

nearly 20 cm. in every metre (1 in 5). The Breguet aeroplane on which Tables I. and II. were based, weighing 600 kg., would assume at different angles (or lift coefficients) the gliding angles shown in Table III.

TABLE III.

Angle corre- Lift. spending to the Lift.	Speed.	Propeller- Thrust in Horizontal Flight.	Gliding Angle Weight (600kg.) divided by figures in col. 5.
m/s.	km/li.
1 2	3 4	5	6
0-02 0°	31-6	113-6	162	0-270
I
0-03 | 2°	25-8	92-8	112	0-187
0-04 ! 4°	22-3	80-3	96	0-160
		|
6i°	20-0	72-0 91	0-151
0-06 10°	18-2	65-6	9f>	0-158
0-066 15°	17-4	62-6	136	0-22C

It will now be seen that the best gliding angle is obtained when the angle of incidence is the same as the optimum

FLIGHT IN STILL AIR 27

angle of the aeroplane. The latter, therefore, is the best from the gliding point of view, so far as the length of the glide is concerned.

In fig. 10, starting from a point 0, are drawn dotted lines corresponding to the gliding angle given in column 6 of Table III., and on these lines are marked off distances proportional to the speed values set out in columns 3 or 4 ; the diagram will then give, if the points are connected into a curve, the positions assumed, in unit time, by a glider, launched at the various angles from the point 0.

It will be observed in the first place that any given gliding path, such as OA, for instance, cuts the curve at two points, A and B, thus showing that this gliding path could have been traversed by the aeroplane at two different speeds, OA and OB, corresponding to the two different angles of incidence, 1° and 15° in the present case.

Only for the single gliding path OM, corresponding to the smallest gliding slope and the optimum angle of inci- dence, do these two points coincide.

But it is not by following this gliding path that an aero- plane will descend best in the vertical sense during a given period of time ; for this it will only do by following the path OE corresponding to the highest point on the curve, and the angle of incidence to be adopted to achieve this result is none other than the economical angle. But the difference in the rate of fall is only slight for the example in question.

It will be noted that as the angle of incidence diminishes, the gliding angle rapidly becomes steeper. If the curve were extended so as to take in very small angles of incidence, it would be found that at a lift coefficient of 0-015 the gliding path would already have become very steep, that this steepness would increase very rapidly for the coefficient 0-010, and that at 0-005 it approached a headlong fall. The fall, in fact, must become vertical when the lift disappears, that is, when the plane meets the air along its imaginary chord.

28 FLIGHT WITHOUT FORMULA

FLIGHT IN STILL AIR 29

In these conditions, a slight variation in the lift there- fore brings about a very large alteration in the gliding angle, and this effect is the more intense the smaller the lift coefficient. The glide becomes a dive. Hence it is clear that this is another danger of adopting a low lift coefficient.

This brief discussion on gliding flight, interesting enough in itself, was necessary to a proper understanding of the part played by power in the horizontal flight of an aero- plane, for we can now regard the latter in the light of a glide in which the gliding path has been artificially raised.

And this raising of the gliding path is due to the power derived from the propelling plant.

This will be better understood if we assume that, during the course of a glide, the pilot started up his engine again without altering the position of the elevator, so that the planes remained at the same angle as before ; the gliding path would gradually be raised until it attained and even surpassed the horizontal, while the aeroplane (as has been seen) would approximately maintain the same speed throughout.

Hence it may be said that when the angle of incidence remains constant, the speed of an aeroplane is not produced by its motive power, as in the case of all other existing vehicles, since, when the motor is stopped, this speed is maintained.

The function of the power-plant is simply to overcome gravity, to prevent the aeroplane from yielding, as it in- evitably must do in calm air, to the attraction of the earth ; in other words, to govern its vertical flight.

In the case now under consideration, the speed therefore is wholly independent of the power, since, as has been seen, it is entirely determined by the angle of incidence, and if this remains constant, as assumed, any excess of power will simply cause the aeroplane to climb, while a lack of power

30 FLIGHT WITHOUT FORMULA

will cause it to coine down, but without any variation in the speed.

But this must not be taken to imply that the available motive-power cannot be transformed into speed, for such, happily, is not the case. Hitherto the elevator has been assumed to be immovable so that the incidence remained constant.

As a matter of fact, the incidence need only be diminished through the action of the elevator in order to enable the aeroplane to adopt the speed corresponding to the new angle of planes, and in this way to absorb the excess of power without climbing.

Nevertheless — and the point should be insisted upon as it is one of the essential principles of aeroplane flight — the angle of incidence alone determines the speed, which cannot be affected by the power save through the intermediary of the incidence.

Hitherto we have constantly alluded to the different speeds at which an aeroplane can fly, as if, in practice, pilots were able to drive their machines at almost any speed they desired. In actual fact, a given aeroplane usually only flies at a single speed, so that we are in the habit of referring to the X biplane as a 70 km. per hour machine, or of stating that the Y monoplane does 100 km. per hour. This is simply because up to now, and with very few exceptions, pilots run their engines at their normal number of revolutions. In these conditions it is evident that the useful power furnished by the propelling plant determines the incidence, and hence the

Thus, referring once again to Table II., it will be seen for example that, if the Breguet biplane receives 29 h.p. in useful power from its propelling plant, the pilot, in order to maintain horizontal flight, will have to manipulate his elevator until the incidence of the planes is approxi- mately 4°, which corresponds to the lift 0-040.

The speed, then, would only be about 80 km. per hour.

FLIGHT IN STILL AIR 31

Experience teaches the pilot to find the correct position of the elevator to maintain horizontal flight. Should the engine run irregularly, and if the aeroplane is to maintain its horizontal flight, the elevator must be slightly actuated in order to correct this disturbing influence.

Horizontal flight, therefore, implies a constant mainten- ance of equilibrium, whence the designation equilibrator, which is often applied to the elevator, derives full justifi- cation.

But if the engine is running normally, the incidence, and consequently the speed, of an aeroplane remain practically constant, and these constitute its normal incidence and speed.

Generally the engine is running at full power during flight, and so in the ordinary course of events the normal speed of an aeroplane is the highest it can attain.

But there is a growing tendency among pilots to reserve a portion of the power which the engine is capable of developing, and to throttle down in normal flight. In this case the reserve of power available may be saved for an emergency, and be used — the case will be dealt with hereafter — for climbing rapidly, or to assume a higher speed for the time being. In this case the normal speed is, of course, no longer the highest possible speed.

In the example already considered, the Breguet biplane would fly at about 80 km. per hour, if it possessed useful power amounting to 29 h.p.

But by throttling down the engine so that it normally only produced a reduced useful power equivalent to 24 h.p., the normal speed of the machine, according to Table II., would only be 72 km. per hour (the normal incidence being 6|° and the lift 0-050).

The pilot would therefore have at his disposal a surplus of power amounting to 5 h.p., which he could use, by, opening the throttle, either for climbing or for temporarily increasing his speed to 80 km. per hour.

32 FLIGHT WITHOUT FORMULA

Although, therefore, an aeroplane usually only flies at one speed, which we call its normal speed, it can perfectly well fly at other speeds, as was shown in Chapter I. But, in order to obtain this result, it is essential that on each occasion the engine should be made to develop the precise amount of power required by the speed at which it is desired to fly.

Speed variation can therefore only be achieved by simultaneously varying the incidence and the power, or, in practice, by operating the elevator and the throttle together. This may be accomplished with greater or less ease according to the type of motor in use, but certain pilots practise it most cleverly and succeed in achieving a very notable speed variation, which i? of great importance, especially in the case of high-speed aeroplanes, at the moment of alighting.

As has already been explained, the horizontal flight of an aeroplane may be considered in the light of gliding flight with the gliding angle artificially raised. From this point of view it is possible to calculate in another way the power required for horizontal flight.

For instance, if we know that an aeroplane of a given weight, such as 600 kg., has, for a given incidence, a glid- ing angle of 16 cm. per metre (approximately 1 in 6) at which its speed is 22-3 m. per second, we conclude that in 1 second it descended 0-16 x 22-3=3-58 m. Hence, in order to overcome its descent and to preserve its hori- zontal flight, it would be necessary to expend the useful power required to raise a weight of 600 kg. to a height of 3-58 m. in 1 second. Since 1 h.p. is the unit required to raise a weight of 75 kg. to a height of 1 m. in 1 second,

the desired useful power = — —about 29 h.p. This,

as a matter of fact, is the amount given by Table II. for the Breguet biplane which complies with the conditions given.

In order to find the useful power required for the

FLIGHT IN STILL AIR 33

horizontal flight of an aeroplane flying at a given incidence, and hence at a given speed, multiply the weight of the machine by this speed and by the gliding angle corresponding to the incidence, and divide by 75.

By a similar method one may easily calculate the useful power required to convert horizontal flight into a climb at any angle.

Thus, if the aeroplane already referred to had to climb, always at the same speed of 22-3 m. per second, at an angle of 5 cm. per metre (1 in 20), it would be necessary to expend the additional power

0-05x600x22-3 ,

=about 9 h.p.

75

Of course, this expenditure of surplus power would be greater the smaller the efficiency of the propeller, and would be 12 h.p. for 75 per cent, efficiency, and 18 h.p. for 50 per cent, efficiency.

Clearly, this method of making an aeroplane climb — by increasing the motive power — can only be resorted to if there is a surplus of power available, that is, if the engine is not normally running at full power, which until now is the exception.

For this reason, when, as is generally the case, the engine is running at full power, climbing is effected in a much simpler manner, which consists in increasing the angle of incidence of the planes by means of the elevator.

Let us once more take our Breguet biplane which, with motor working at full power, flies at a normal speed of 22-3 m. per second (80-3 km. per hour) at 4° incidence (or a lift coefficient of 0-040). The useful power needed to achieve this speed (see Table II.) is 29 h.p.

Assume that, by means of his elevator, the pilot increases the angle of incidence to 10° (lift coefficient 0-060). Since horizontal flight at this incidence, which must inevitably reduce the speed to 18-2 m. per second or 65-6 km. per hour, would only require 23 h.p., there will be an ex-

3

34 FLIGHT WITHOUT FORMULAE

cess of power amounting to 6 h.p.,* and the aeroplane will rise.

The climbing angle can be calculated with great ease. The method is just the converse of the one we have just employed, and thus consists in dividing 6x75 (representing the surplus power) by 600 x20 (weight multiplied by speed), which gives an angle of 3*75 cm. per metre (1 in 27 about).

This climbing rate may not appear very great ; still, for a speed of 18-2 m. per second, it corresponds to a climb of 68 cm. per second=41 m. per minute=410 m. in 10 minutes, which is, at all events, appreciable.

The aeroplane, therefore, may be made to climb or to descend by the operation of the elevator by the pilot.

More especially is the elevator used for starting. In this case the elevator is placed in a position corresponding to a very slight incidence of the main planes, so that these offer very little resistance to forward motion when the motor is started and the machine begins to run along the ground. As soon as the rolling speed is deemed sufficient, the elevator is moved to a considerable angle, which causes the planes to assume a fairly high incidence, and the aero- plane rises from the ground.

* This is not strictly correct, since, as will be seen hereafter, the propeller efficiency varies to some extent with the speed of the aeroplane ; still, we shall not make a grievous error in assuming that the efficiency remains the same.

CHAPTER III

FLIGHT IN STILL AIR

POWER (concluded]

THE second chapter was mainly devoted to explaining how one may calculate the useful power required for horizontal flight, at the various angles of incidence and at the different lift coefficients — in other words, at the various speeds of a given aeroplane.

In addition, gliding flight has been briefly touched on, and has served to show the precise manner in which the power employed affects the speed of the aeroplane.

In the present chapter this discussion will be completed ; it will be devoted to finding the best way of employing the available power to obtain speed. Incidentally, we shall have occasion to deal briefly with the limits of speed which the aeroplane as we know it to-day seems capable of attaining.

It has been shown that the flight of a given aeroplane requires a minimum useful power, and that this is only possible when the angle of incidence is that which we have termed the economical angle.

The power would therefore be turned to the best account, having regard merely to the sustentation of the aeroplane, by making it fly normally at its economical angle.

But, on the other hand, this method is most defective from the point of view of speed, for as fig. 6 (Chapter II.) clearly shows, when the machine flies at its economical angle, a very slight increase in power will increase the

35

36 FLIGHT WITHOUT FORMULA

speed to a considerable extent. Besides, the method in question would be worthless from a practical point of view, since it is evident that an aeroplane flying under these conditions would be endangered by the slightest failure of its engine.

Such, in fact, was the case with the first aeroplanes which actually rose from the ground ; they flew " without a margin," to use an expressive term. And even to-day the same is true of machines whose motor is running badly : in such a case the only thing to be done is to land as soon as possible, since the aeroplane will scarcely respond to the controls.

The other characteristic value of the angle of incidence referred to in Chapter II., there called the optimum angle, corresponds to the least value of the ratio between the propeller-thrust and the weight of the aeroplane, or to its equivalent — the best gliding angle.

For the best utilisation of the power in order to obtain speed, which alone concerns us for the moment, there is a distinct advantage attached to the use of the optimum angle for the normal incidence of the machine ; Colonel Renard, indeed, long ago pointed out that by using the optimum angle for normal flight in preference to the economical angle, one obtained 32 per cent, increase in speed for an increase in power amounting to 13 per cent. only.

In any case, when the incidence is optimum the ratio between the speed and the useful power required to obtain it is largest. This is easily explained by reference to Chapter II., which showed that the useful power required for horizontal flight at a given incidence is proportional to the speed multiplied by the gliding angle of the aeroplane at the same incidence.

When the gliding angle is least (i.e. flattest), that is, when the incidence is that of the optimum angle, the ratio of power to speed is also smallest, and hence the ratio of speed to maximum power.

It would therefore appear that by using the optimum

FLIGHT IN STILL AIR 37

angle as the normal incidence we would obtain the best results from the point of view with which we are at present concerned, which is that of the most profitable utilisation of the power to produce speed. This, in fact, is generally accepted as the truth, and in his scale model experiments M. Eiffel always recorded this important value of the angle of incidence, together with the corresponding flattest gliding angle.

Nevertheless we are not prepared to accept as inevitably true that the optimum angle is necessarily the most ad- vantageous for flight, so far as the transmutation of power into speed is concerned. This will now be shown by approaching the question in a different manner, and by finding the best conditions under which a given speed can be attained.

The power required for flight is proportional, as has been shown, to the propeller-thrust multiplied by the speed. Hence, on comparing different aeroplanes flying at the same speed, it will be found that the values of the power ex- pended to maintain flight will have the same relation to one another as the corresponding values of the propeller-thrust.

If we assume that the detrimental surface of each one of these aeroplanes is identical, the head-resistance will be the same in each case, since it is proportional to the detri- mental surface multiplied by the square of the speed (which is identical in every case).

It follows that the speed in question will be attained most economically by the aeroplane whose planes exert the least drag. Now, it was shown in Chapter II. that the drag of the wings of an aeroplane is a fraction of the weight of the machine equal to the ratio between the drag coefficient and the lift coefficient corresponding to the incidence at which flight is made.

If we assume, therefore, that the weight of each aero- plane is identical, it follows that the best results are given by that machine whose planes in normal flight have the smallest drag-to-lift ratio.

38 FLIGHT WITHOUT FORMULA

Reference to the polar diagrams (Chapter I., figs. 1, 2, 3, and 4) shows that the minimum drag-to-lift ratio occurs at the angle of incidence corresponding to the point on the curve where a straight line rotated about the centre 0-00 comes into contact with the curve. This angle of incidence is beyond all question, for any aeroplane provided with planes of the types under consideration, the most profitable from our point of view ; this angle, in other words, is that at which an aeroplane of given weight can fly at a given speed for the least expenditure of power, and this for any weight and speed. Hence this is the angle at which an aeroplane possessing one of these wing sections should always fly in theory. Accordingly, it may be termed the be t angle of incidence, and the corresponding lift coefficient the best lift coefficient.

The value of the best incidence only depends on the wing section, but it is always smaller than the optimum angle, which in its turn depends not only on the wing section but also on the ratio of the detrimental surface to the plane area.

A straight line rotated from the centre 0-00 in figs. 2, 3, and 4 indicates that the best lift coefficients for M. Farman, Breguet, and Bleriot XI. plane sections are respectively 0-017, 0-035, and 0-047, corresponding to the best angles of incidence 1|°, 2£°, and 6°. These values can only be de- termined with some difficulty, however, since the curves are so nearly straight at these points that the rotating line would come into contact with the curves for some distance and not at one precise point alone.

On the other hand, it is evident that the drag-to-lift ratio only varies very slightly for a series of angles of incidence, the range depending on the particular plane section, so that one is justified in saying that each type of wing pos- sesses not only one best incidence and one best lift, but several good incidences and good lifts.

Thus, for the Maurice Farman section, the good lifts lie between 0-010 and 0-025 approximately, and the corre-

FLIGHT IN STILL AIR 39

spending good incidences extend from 1° to 4°, while the drag-to-lift ratio between these limits remains practically constant at 0-065.

For the Breguet wing, the good lifts are between 0-030 and 0-045, the good incidences between 3° and 6°, and the drag-to-lift ratio remains about 0-08.

Lastly, for the Bleriot XI. the same values read as follows : 0-030 and 0-055, 3° and 6°, and about 0-105.

Even at this point it becomes evident that the use of .slightly cambered wings is the more suitable for flight with a low lift coefficient, and that for a large lift a heavily cambered wing is preferable.

If the optimum angle of an aeroplane, which depends, as already shown, on the ratio between the detrimental surface and the plane area, is included within the limits of the good incidences, its use as the normal angle of in- cidence remains as advantageous as that of any other " good " incidence. But if it is not included,* flight at the optimum angle would require, in theory at all events, a greater expenditure of power than would be required under similar conditions if flight took place at any of the good incidences.

This shows that the optimum angle is not necessarily that at which an aeroplane should fly normally in order to use the power most advantageously.

To sum up : the normal speed should always correspond to a " good " angle of incidence.

Should this not be the case in fact, it would be possible to design an aeroplane which, for the same weight and detrimental surface as the one under consideration, could achieve an equal speed for a smaller expenditure of power.

A concrete example will render these considerations clearer.

In Table II. (Chapter II.) there was set out the variation

* This would be possible more particularly in the case of aeroplanes with very slightly cambered planes and small wing area and considerable detrimental surface.

40 FLIGHT WITHOUT FORMULA

of the useful power required for the horizontal flight of a Breguet aeroplane weighing 600 kg., with a plane area of 30 sq. m. and a detrimental surface of 1-20 sq. m., according to its speed.

Let us assume that the useful power — 24 h. p. —developed by the propeller makes the aeroplane fly normally at 0-050 lift, or at its optimum incidence. The speed will then be 72 km. per hour or 20 m. per second. This lift coefficient 0-050, be it noted, is slightly greater than the highest of the good incidences peculiar to the Breguet section.

Now let us take another aeroplane of the same type, also weighing 600 kg. and with the same detrimental surface of 1-20 sq. m., but with 40 sq. m. plane area, which should still fly at the same speed of 20 m. per second.

The lift coefficient may be obtained (cf. Chapter I.) by dividing the loading of the planes (15 kg.) by the square of the speed in metres per second (400), which gives 0-0375. Now this is one of the good lift coefficients of the Breguet plane. In these conditions, therefore, the drag-to-lift ratio will assume the constant value of about 0-08 common to all good incidences.

It follows that the drag of the planes will be equal to the weight, 600 kg. x 0-08=48 kg.

The head-resistance, on the other hand, will remain the same as in the original aeroplane whose speed was 72 km. per hour, since head-resistance is dependent simply on the amount of detrimental surface and on the speed (neither of which undergoes any change). The head-resistance, there- fore (cf. Chapter II.), equals 38 kg.

The propeller-thrust, equal to the sum of head-resistance and drag of the planes, will be 86 kg., and the useful power required for flight =

Thrust (86) XqeedjjO^ ^

75

The figure thus obtained is less than the 24 h.p. of useful

FLIGHT IN STILL AIR 41

power required to make the aeroplane first considered fly at 72 km. per hour.

Therefore, in theory at all events, the optimum angle is not necessarily the most advantageous from the point of view of the least expenditure of power to obtain speed. But in practice the small saving in power would probably be neutralised owing to the difficulty of constructing two aeroplanes of the same type with a plane area of 30 and 40 sq. m. respectively without increasing the weight and the detrimental surface of the latter. Hence the advantage dealt with would appear to be purely a theoretical one in the present case.

But this would not be so with an aeroplane whose normal angle of incidence was smaller than the good incidences belonging to its particular plane section. For instance, let us assume that the propeller of the Breguet aeroplane (vide Table II.) furnishes normally 68 useful h.p., which would give the machine a speed of 113-6 km. per hour or 31-6 m. per second, at the lift 0-020, which is less than the good lifts for this plane section.

Now take another Breguet aeroplane of the same weight and detrimental surface, but with a plane area of only 20 sq. m. Calculating as before, it will be found that in order to achieve a speed of 113-6 km. per hour, this machine Avould have to fly with a lift of 0-030, which is one of the good lifts, and that useful power amounting to only 60 h.p. would be sufficient to effect the purpose. This time the advantage of using a good incidence as the normal angle is clearly apparent.

As a matter of fact, in practice the advantage would probably be even more considerable, since a machine with 20 sq. m. plane area would probably be lighter and have less detrimental surface than a 30 sq. m. machine.

Care should therefore be taken that the normal angle of an aeroplane is included among the good incidences belonging to its plane section, and, above all, that it is not smaller than the good incidences.

42 PLIGHT WITHOUT FORMULA

This manner of considering good incidences and lifts provides a solution of the following problem which was referred to in Chapter I. :

Since there are only two means of increasing the speed of an aeroplane — either by increasing the plane loading or by reducing the lift coefficient — which of these is the more economical ?

To begin with, the question will be examined from a theoretical point of view, by assuming that the adoption of either means will have the same effect in each case on the weight and the detrimental surface, since the values of these must be supposed to remain the same in the various machines to enable our usual method of calculation to be applied.

This being so, it will be readily seen that as long as the normal lift remains one of the good lifts, both means of increasing the speed are equivalent as far as the expenditure of useful power is concerned.

On the one hand, since the drag-to-lift ratio retains approximately the same value for all good lifts, the drag of the planes will remain for every angle of incidence a constant fraction of the weight, which is assumed to be in- variable. On the other hand, at the speed it is desired to attain, the head-resistance, proportional to the detrimental surface, which is also assumed to be invariable, will remain the same in both cases. Consequently, the propeller-thrust, equal to the sum of the two resistances (drag of the planes 4-head-resistance), and hence the useful power, will retain the same value by whichever of the two methods the increase in speed has been obtained.

But if the lift had already been reduced to the smallest of the good lift values, and it was still desired to increase the speed, the most profitable manner of doing this would be to increase the loading by reducing the plane area. So much for the theoretical aspect of the problem.

Purely practical considerations strengthen these theor- etical conclusions, in so far as they clearly prove the ad-

FLIGHT IN STILL AIR 43

vantage of increasing the speed by the reduction of plane area, even where the lift remains one of the good lift values.

Indeed, in practice the two methods are no longer equiva- lent in the latter case, since, as already mentioned, the reduction of the wing area is usually accompanied by a decrease in the weight and detrimental surface.

Generally speaking, it is therefore preferable to take the highest rather than the lowest of the good lifts as the normal angle of incidence, and this conclusion tallies, moreover, with that arising from the danger of flying at a very low lift. Finally, the normal angle would thus remain in the neighbourhood of the optimum angle, which is an excellent point so far as a flat gliding angle is concerned.*

Obviously, the advantage of the method of increasing the speed by reducing the plane area over that consisting in reducing the lift becomes greater still in the case where the latter method, if applied, would lead to the lift being less than any of the good lift values.

The disadvantage of greatly reducing the plane area to obtain fast machines is the heavy loading which it entails and the lessening of the gliding qualities. The best practical solution of the whole problem would therefore appear to consist in a judicious compromise between these two methods.

As usual, a concrete example will aid the explanation given above.

Let the Breguet aeroplane already referred to be supposed to fly at a speed of 92-8 km. per hour with a lift of 0-030, which is the lowest of its good lift values. Table II. shows that this would require 38 h.p.

Another machine of the same type, and having the same weight and detrimental surface, but with an area of only 20 sq. m. (instead of 30), in order to attain the same speed

* Chapter X. will show that this conclusion is strengthened still further by the effect of wind on the aeroplane.

44 FLIGHT WITHOUT FORMULA

would have to fly at 0-040 lift, which is also one of the good lift values.

The necessary calculations would show that the latter machine, like the former, would also require 38 h.p. This is readily explicable on the score that the drag of the planes is 0-08 of the weight, or 48 kg., while the head-resistance also remains constant and equal to 64 kg. (Table II.).

In theory, therefore, there is nothing to choose between either solution. But in practice the latter is preferable, since the 20 sq. m. machine would in all likelihood be lighter and possess less detrimental surface.

But if a speed of 113-6 km. per hour were to be attained, the 20 sq. m. aeroplane has a distinct advantage both in theory, and even more in practice, for the machine with 30 sq. m. area would have to fly at 0-020 lift, which is lower than the good lift values belonging to the Breguet plane section, which would, as already shown, require useful power amounting to 68 h.p., whereas 60 h.p. would suffice to maintain the smaller machine in flight at the same speed.

We have already set forth the good lift values belonging to the Maurice Farman, Breguet, and Bleriot XI. plane sections, and the corresponding values of the drag-to-lift ratio or, its equivalent, the ratio of the drag of the planes to the weight of the machine. .

Reference to these values has already shown that slightly cambered planes are undoubtedly more economical for low lift values, which are necessary for the attainment of high speeds, especially in the case of lightly loaded planes, as in some biplanes.

But the good lift values of very flat planes are usually very low — from 0-010 to 0-025 in the case of the Maurice Farman — which greatly restricts the use of these values, since, as already stated, it is doubtful whether hitherto an aeroplane has flown at a lower lift value than 0-020.

The advantages and disadvantages of these three wing sections, from the point of view at issue, will be more readily

FLIGHT IN STILL AIR

45

BL	?RIOT 0^	n°XI
		s \
Bf	EGUE	TO^	\
			5
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M.I	'ARM*	I/V	\
			1
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				1
				I
				!
				1 o1 • 1 9
				, i 4 !
				A

O.Q8

0.07

0.08

seen by plotting their polar curves in one diagram, as shown in fig. 11.

The Breguet and Maurice Farman curves intersect at a point corresponding to the lift value 0-030, whence we may conclude that for all lift values lower than this, the Maurice Farman section is the better,* but for all lift values higher than 0-032 (which at present are more usual), the Breguet wing has a distinct advantage. In the same way, the Maurice Farman is better than the Bleriot XI. for lift values below 0-042, whereas the latter is better for all higher lift values.

Finally, the Bleriot XI. only becomes superior to the Breguet for lift values in excess of 0-065, which are very high indeed, and little used owing to the fact that they correspond to angles in the neighbour- hood of the economical angle.

To apply these various considerations, we will now proceed to fix the best con- ditions in which to obtain

A 0.04

0.03

0.02

0.01

0.02 Drag.

0.01

o'uo

o.oq

FIG. 11

a speed of 160 km. per hour or about 44-5 m. per second, which appears to be the highest speed which it seems at present possible to * Since it has a smaller drag for the same lift.

46 FLIGHT WITHOUT FORMULA

reach,* that is, by assuming it to be possible to have a loading of 40 kg. per sq. m. of surface and to fly at a lift- value of 0-020.

In laying down this limit to the speed of flight we also stated our belief that, in order to enable it to be attained, engines developing from 120 to 130 effective h.p. would have to be employed.

This opinion was founded on the results of M. Eiffel's experiments, from which it was concluded that an aeroplane to attain this speed would have to possess a detrimental surface of no more than 0-75 sq. m.

Now, the last two Aeronautical Salons, those of 1911 and 1912, have shown a very clearly marked tendency among constructors to reduce all passive resistance to the lowest possible point, especially in high-speed machines, and it would appear that in this direction considerable progress has been and is being made.

One machine in particular, the Paulhan-Tatin " Torpille," specially designed with this point in view, is worthy of notice.

Its designer, the late M. Tatin, estimated the detrimental surface of this aeroplane at no more than 0-26 sq. m., and its resistance must in fact have been very low, since it had the fair-shaped lines of a bird, every part of the structure capable of setting up resistance being enclosed in a shell-like hull from which only the landing wheels, reduced to the utmost verge of simplicity, projected.

Taking into account the slightly less favourable figures obtained by M. Eiffel from experiments with a scale model, the detrimental surface of the " Torpille " may be estimated at 0-30 sq. m.

According to information given by M. Tatin himself, the weight of this monoplane was 450 kg., and its plane area 12-5 sq. m.

* It should, however, be remembered that this limit has actual^ been exceeded, with a loading of 44 kg. per sq. m. and a lift value of slightly less than 0*020. See also Translator's note on p. 12.

FLIGHT IN STILL AIR 47

Let us assume that the planes, which were only very slightly cambered, were about equivalent to those of the Maurice Farman, and that they flew at a good lift coefficient. In that case the drag of the planes would be equal to 0-065 of the weight of the machine, or to 29-5 kg.

On the other hand, at the speed of 44-5 m. a second, the head-resistance =

n ffi • Detrimental Square of

Coefficient. ^^ ^ s^

0-08 x 0-3 X 1980 - 47-5 kg.

The propeller-thrust, consequently, the sum of both resistances, would =17 kg.

The useful power required would thus=

77x44-5 ,

=about 45 h.p.

75

Propeller efficiency in this case must have been exception- ally high (as will be seen hereafter), and was probably in the region of 80 per cent.

The engine -power required to give the " Aero-Torpille " a

A£*

speed of 160 km. per hour must therefore have been — =

0*8

57 h.p., or approximately 60 h.p.

M. Tatin considered that he could obtain the same result with even less motive-power, and that some 45 h.p. would suffice. If this proves to be the case, the detrimental surface of the aeroplane would have to be less than 0-30 sq. m. and the propeller efficiency even higher than 80 per cent., or else — and this was M. Tatin's own opinion — the coefficients derived from experiments with small scale models must be increased for full-size machines, their value possibly depending in some degree on the speed.*

* No proof, as a matter of fact, was possible owing to the short life of the machine. But the results obtained from other machines in which stream-lining had been carried out to an unusual degree, such as the Deperdussin " monocoque " — which, with an engine of 85-90 effective h.p., only achieved 163 km. per hour — would appear to show that the

48 FLIGHT WITHOUT FORMULA

It should also be noted that, in order to attain 160 km. per hour, the Tatin " Torpille " would have to fly at a lift coefficient equal to

36 (loading) =0-018

1980 (square of the speed)

Perhaps it will seem strange that simply by estimating the value of the detrimental surface at 0-30 instead of the previous estimate of 0-75, the motive power required for flight at 160 km. per hour should have been reduced by one-half. Yet there is no need for surprise ; for if the method for calculating the useful power necessary for horizontal flight (set forth in Chapter II., and since applied more than once) is carefully examined, it becomes evident that, whereas that portion of the power required only for lifting remains proportional to the speed, the remaining portion, used to overcome all passive resistance, is propor- tional to the cube of the speed.

For this reason it is of such great importance to cut down the detrimental surface hi designing a high-speed machine.

Thus, in the present case, of the 46 h.p. available, only 18 h.p. are required to lift the machine. The remain- ing 28 h.p., therefore, are necessary to overcome passive resistance.

Had the detrimental surface been 0-75 sq. m. instead of 0-30, the useful power absorbed in overcoming passive resistance would have been

Q.75x28=7() h mgtead 2g 0-30

To complete our examination of the high-speed aeroplane, Table IV. has been drawn up, and includes the values of the useful power required on the one hand for the flight of a Maurice Farman plane at a good incidence, and weighing

estimate of 0'30 sq. m. for the detrimental surface was too low, a con- clusion supported by M. Eiffel's experiments.

It is doubtful whether an aeroplane has yet been built with a detri- mental surface of much less than half a square metre.

FLIGHT IN STILL AIR

49

1 ton (metric), and on the other for driving through air a detrimental surface of 1 sq. m. at speeds from to 200 km. per hour.

TABLE IV.

the 150

Speed.		lllL	II. J^s |3*g*-i	|||.si
Km.	Metres	Drag of Planes (kg.) ; per aeroplane ton.	-sll^l	1151II fsflll	Ijjn
per hr.	per see.	"* £ ^ >> "	3 f *" ^oo g	^ f, 2 -S ^
		g * ^*>	(2«S ^6	}S «se^ 8
1	2	3	4	5	6
150	41-6	65	36	138	76
160	44-4^	•slsf	38	157	93
170	47-2	l?|ol	41	178	112
180 190	50-0 52-8		43 46	200 232	132 157
200	55-6,	-** K 5	48	248	184
		•^ o

According to this Table, an aeroplane weighing 500 kg., and possessing, as we supposed in the case of the Tatin " Torpille," a detrimental surface of 0-30 sq. m., would re- quire a useful power of about 80 h.p. to attain a speed of 200 km. per hour. This high speed could therefore be achieved with a power-plant consisting of a 100-h.p. motor and a propeller of 80 per cent, efficiency. It could only be obtained — just as the " Torpille " could only achieve 160 km. per hour at a lift coefficient of 0-018 — with a plane loading of about 56 kg. per sq. m. Consequently, the

area of the planes would be only -=^- = 9 sq. m.

OD

If the theoretical qualities of design of machines of the " Torpille " type are borne out by practice * our present

* But, according to what has already been said, this does not seem to be the case. Hence, a speed of 200 km. per hour is not likely to be

50 FLIGHT WITHOUT FORMULA

motors would appear to be sufficient to give them a speed of 200 km. per hour. But this would necessitate a very heavy loading and a lift coefficient much lower than any hitherto employed — a proceeding which, as we have seen, is not without danger. Moreover, one cannot but be uneasy at the thought of a machine weighing perhaps 500 or 600 kg. alighting at this speed.

This, beyond all manner of doubt, is the main obstacle which the high-speed aeroplane will have to overcome, and this it can only do by possessing speed variation to an exceptional degree. We will return to this aspect of the matter subsequently.

To-day an aeroplane, weighing with full load a certain weight and equipped with an engine giving a certain power, in practice flies horizontally at a given speed.

These three factors, weight, speed, and power, are always met with whatever the vehicle of locomotion under con- sideration, and their combination enables us to determine as the most efficient from a mechanical point of view that vehicle or machine which requires the least power to attain, for a constant weight, the same speed.

Hence, what we may term the, mechanical efficiency of an aeroplane may be measured through its weight multiplied by its normal speed and divided by the motive- power.

If the speed is given in metres per second and the power in h.p., this quotient must be divided by 75.

RULE. — The mechanical efficiency of an aeroplane is obtained by dividing its weight multiplied by its normal speed (in metres per second) by 75 times the power, or, what is the same thing, by dividing by 270 times the power the product of the weight multiplied by the speed in kilometres per hour.

EXAMPLE. — An aeroplane weighing 950 kg., and driven

attained with a 100-h.p. motor. Whether an engine developing 140 h.p. or more will succeed in this can only be shown by the future, and perhaps at no distant date. See footnote, p. 12.

FLIGHT IN STILL AIR 51

by a, lOQ-h.p. engine, flies at a normal speed of 117 km. per hour. What is its mechanical efficiency ?

950x117

Reference to what has already been said will show that mechanical efficiency is also expressed by the propeller efficiency divided by the gliding angle corresponding to normal incidence. This is due to the fact that, firstly, the useful power required for horizontal flight is the 75th part of the weight multiplied by the speed and the normal gliding angle, and, secondly, because the motive power is obtained by dividing the useful power by the propeller efficiency. Accordingly, a machine with a propeller efficiency of 70 per cent., and with a normal gliding angle of 0-17,

would have a mechanical efficiency—— — =4-12.

This conception of mechanical efficiency enables us to judge an aeroplane as a whole from its practical flying performances without having recourse to the propeller efficiency and the normal gliding angle, which are difficult to measure with any accuracy.

Even yesterday a machine possessing mechanical efficiency superior to 4 was still, aerodynamically considered, an excellent aeroplane. But the progress manifest in the last Salon entitles us, and with confidence, to be more exacting in the future.

Hence, the average mechanical efficiency of the ordinary run of aeroplanes enables us in some measure to fix definite periods in the history of aviation. In 1910, for instance, the mean mechanical efficiency was roughly 3-33, on which we based the statement contained in a previous work that, in practice, 1 h.p. transports 250 kg. in the case of an average aeroplane at 1 m. per second.

This rule, which obviously only yielded approximate results, could be applied both quickly and easily, and enabled one, for instance, to form a very fair idea of the results

52 FLIGHT WITHOUT FORMULA

that would be attained in the Military Trials of 1911. In fact, according to the rules of this competition, the aero- planes would have to weigh on an average 900 kg. To

give them a speed of 70 km. per hour or — - m. per second,

3'6

c • i A • 90° v 70

for instance, the rule quoted gives — — x^-^-

250 o'b

But 250 X 3-6 remains the denominator whatever the speed it is desired to attain, and is exactly equal to 900, the weight of the aeroplane. From this, one deduced that in this case the power required in h.p. was equivalent to the speed in kilometres per hour : —

70 km. per hour 70 h.p.

80 „ ..... 80 h.p. 100 ,, 100 h.p.

If, on the other hand, certain machines during these trials, driven by engines developing less than 100 effective h.p., flew at over 100 km. per hour, this was due simply to their mechanical efficiency being better than the 3-33 which obtained in 1910, and was already too low for 1911.

At the present day, therefore, accepting 4 as the average mechanical efficiency, the practical rule given above should be modified as follows : —

RULE. — 1 h.p. transports 300 kg. of an average aeroplane at 1 m. per second.

CHAPTER IV

FLIGHT IN STILL AIR THE POWER-PLANT

BOTH this chapter and the next will be devoted to the power-plant of the aeroplane as it is in use at the present time. This will entail an even closer consideration of the part played by the motive-power in horizontal and oblique flight, and will finally lead to several important conclusions concerning the variable-speed aeroplane and the solution of the problem of speed variation.

The power-plant of an aeroplane consists in every case of an internal combustion motor and one or more propellers. Since the present work is mainly theoretical, no description of aviation motors will be attempted, and only those of their properties will be dealt with which affect the working of the propeller.

Besides, the motor works on principles which are beyond the realm of aerodynamics, so that from our point of view its study has only a minor interest. It forms, it is true, an essential auxiliary of the aeroplane, but only an auxiliary. If it is not yet perfectly reliable, there is no doubt that it will be in a few years, and this quite independently of any progress in the science of aerodynamics.

Deeply interesting, on the other hand, are the problems relating to the aeroplane itself, or to that mysterious contrivance which, as it were, screws itself into the air and transmutes into thrust the power developed by the engine.

54 FLIGHT WITHOUT FORMULA

The power developed by an internal combustion engine varies with the number of revolutions at which the resist- ance it encounters enables it to turn. There is a generally recognised ratio between the power developed and the speed of revolution.

Thus, if a motor, normally developing 50 h.p. at 1200 revolutions per minute, only turns at 960 revolutions per

50 X 960 minute, it will develop no more than =40 h.p.

The rule, however, is not wholly accurate, and the variation of the power developed by a motor with the number of revolutions per minute is more accurately shown in the curve in fig. 12. It should be clearly understood that the curve only relates to a motor with the throttle fully open, and where the variation in its speed of rotation is only due to the resistance it has to overcome.

For the speed of rotation may be reduced in another manner — by shutting off a portion of the petrol mixture by means of the throttle. The engine then runs "throttled down," which is the usual case with a motor car.

In such a case, if the petrol supply is constant, the curve in fig. 12 grows flatter, with its crest corresponding to a lower speed of rotation the more the throttle is closed and the explosive mixture reduced.

Fig. 13 shows a series of curves which were prepared at my request by the managing director of the Gnome Engine Company ; these represent the variation in power with the speed of rotation of a 50-h.p. engine, normally running at 1200 revolutions per minute, with the throttle closed to a varying extent.

In practice, it is easier to throttle down certain engines than others ; with some it is constantly done, with others it is more difficult.

Even to-day the working of a propeller remains one of the most difficult problems awaiting solution in the whole range of aerodynamics, and the motion, possibly whirling, of the air molecules as they are drawn into the revolving

FLIGHT IN STILL AIR

Horse-Power.

55

& 3 3..

56

FLIGHT WITHOUT FORMULA

Horse-Power. .5 $ § * § J? 3 3 3 ^ Lja__2
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FLIGHT IN STILL AIR 57

propeller has never yet been explained in a manner satis- factory to the dictates of science.

All said and done, the rough method of likening a propeller to a screw seems the most likely to explain the results obtained from experiments with propellers.

The pitch of a screw is the distance it advances in one revolution in a solid body. The term may be applied in a similar capacity to a propeller. The pitch of a propeller, therefore, is the distance it would travel forwards during one revolution if it could be made to penetrate a solid body. But a propeller obtains its thrust from the reaction of an elusive tenuous fluid. Clearly, therefore, it will not travel forward as great a distance for each revolution as it would if screwing itself into a solid.

The distance of its forward travel is consequently always smaller than the pitch, and the difference is known as the slip. But, contrary to an opinion which is often held, this slip should not be as small as possible, or even be altogether eliminated, for the propeller to work under the best con- ditions.

Without attempting to lay down precisely the phenomena produced in the working of this mysterious contrivance, we may readily assume that at every point the blade meets the air, or " bites " into it, at a certain angle depending, among other things, on the speed of rotation and of forward travel of the blade and of the distance of each point from the axis.

Just as the plane of an aeroplane meeting the air along its chord would produce no lift, so a propeller travelling forward at its pitch speed — that is, without any slip — would meet the air at each point of the blades at no angle of incidence, and consequently would produce no thrust.

The slip and angle of incidence are clearly connected together, and it will be easily understood that a given propeller running at a given number of revolutions will have a best slip, and hence a lest forward travel, just as a given plane has a best angle of incidence.

58 FLIGHT WITHOUT FORMULA

When the propeller rotates without moving forward through the air, as when an aeroplane is held stationary on the ground, it simply acts as a ventilator, throwing the air backwards, and exerts a thrust on the machine to which it is attached. But it produces no useful power, for in mechanics power always connotes motion.

But if the machine were not fixed, as in the case of an aeroplane, and could yield to the thrust of the propeller, it would be driven forward at a certain speed, and the product of this speed multiplied by the thrust and divided by 75 represents the useful power produced by the pro- peller.

On the other hand, in order to make the propeller rotate it must be acted upon by a certain amount of motive power. The relation between the useful power actually developed and the motive power expended is the efficiency of the propeller.

But the conditions under which this is accomplished vary, firstly, with the number of revolutions per minute at which the propeller turns, and secondly, with the speed of its forward travel, so that it will be readily understood that the efficiency of a propeller may vary according to the conditions under which it is used.

Experiments lately conducted — notably by Major Dorand at the military laboratory of Chalais-Meudoii and by M. Eiffel — have shown that the efficiency remains approximately constant so long as the ratio of the forward speed to its speed of revolution, i.e. the forward travel per revolution, remains constant.

For instance, if a propeller is travelling forward at 15 m. per second and revolving at 10 revolutions per second, its efficiency is the same as if it travelled forward at 30 m. per second and revolved at 20 revolutions per second, since in both cases its forward travel per revolution is 1-50 m.

But the propeller efficiency varies with the amount of its forward travel per revolution.

FLIGHT IN STILL AIR 59

Hence, when the propeller revolves attached to a stationary point, as during a bench test, so that its forward travel is zero, its efficiency is also zero, for the only effect of the motive power expended to rotate the propeller is to produce a thrust, which in this instance is exerted upon an immovable body, and therefore is wasted so far as the production of useful power is concerned.

Similarly, when the forward travel of the propeller per revolution is equal to the pitch, and hence when there is no slip, it screws itself into the air like a screw into a solid ; the blades have no angle of incidence, and therefore produce no thrust.*

Between the two values of the forward travel per revolution at which the thrust disappears, there is a value corresponding on the other hand to maximum thrust. This has already been pointed out, and has been termed the best forward travel per revolution.

This shows that the thrust of one and the same propeller may vary from zero to a maximum value obtained with a certain definite value of the forward travel. The variation of the thrust with the forward travel per revolution may be plotted in a curve. A single curve may be drawn to show this variation for a whole family of propellers, geometri- cally similar and only differing one from the other by their diameter.

Experiments, in fact, have shown that such propellers had approximately the same thrust when their forward travel per revolution remained proportional to their diameter.

Thus two propellers of similar type, with diameters measuring respectively 2 and 3m., would give the same thrust if the former travelled 1-2 m. per revolution and

* This could never take place if the vehicle to which the propeller was attached derived its speed solely from the propeller ; it could only occur in practice if motive power from some outside source imparted to the vehicle a greater speed than that obtained from the propeller- thrust alone.

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the latter 1-8 m., since the ratio of forward travel to diameter =0-60.

This has led M. Eiffel to adopt as his variable quantity not the forward travel per revolution, but the ratio of this advance to the diameter, which ratio may be termed reduced forward travel or advance.

Fig. 14, based on his researches, shows the variation in thrust of a family of propellers when the reduced advance assumes a series of gradually increasing values.*

The maximum thrust efficiency (about 65 per cent, in this case) corresponds to a reduced advance value of 0-6.

Reduced Advance. FIG. 14.

Hence a propeller of the type under consideration, with a diameter of 2-5 m., in order to give its highest thrust, would have to have a forward travel of 2-5 xO'6=l-2 m. Consequently, if in normal flight it turned at 1200 revolutions per minute, or 20 revolutions per second, the machine it propelled ought to fly at 1-20x20=24 m. per second.

For all propellers belonging to the same family there exists, therefore, a definite reduced advance which is more

* Actually, M. Eiffel found that for the same value of the reduced advance the thrust was not absolutely constant, but rather that it tended to grow as the number of revolutions of the propeller increased. Accordingly, he drew up a series of curves, but these approximate very closely one to another.

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61

favourable than any other, and may thence be termed the best reduced advance, which enables any of these propellers to produce their maximum thrust.

It has been shown that all geometrically similar pro- pellers— in other words, belonging to the same family — give approximately the same maximum thrust efficiency.

But when the shape of the propeller is changed, this maximum thrust value also varies.

It depends more especially on the ratio between the pitch of the propeller and its diameter, which is known as the pitch ratio.

But, as the value of the highest thrust varies with the pitch ratio, so does that of the best reduced advance corre- sponding to this highest thrust.

In the following Table V., based on Commandant Dorand's researches at the military laboratory of Chalais-Meudon with a particular type of propeller, are shown the values of the maximum thrust and the best reduced advance corresponding to propellers of varying pitch ratio.

TABLE V.

Pitch ratio .	0-5	0-6	0-7	0-8	0-9	1-0	I'M
Maximum thrust efficiency	0-45	0'53	0'61	0-70	0-76	0-80	0-84
Best reduced advance	0"29	0-38	0-47	0-55	0-63	072	0-84

EXAMPLE. — A propeller of the Chalais-Meudon type with 2-5 m. diameter and 2 m. pitch turns at 1200 revolutions per minute.

1. What is the value of its highest thrust efficiency ?

2. What should be the speed of the aeroplane it drives in order to obtain this highest thrust ?

The pitch ratio is — ~=0-8. Table V. immediately solves the first question : the

62 FLIGHT WITHOUT FORMULA

highest thrust efficiency is 0-7. Further, this table shows that to obtain this thrust the reduced advance should =0-55. In other words, the speed of the machine divided by 50 (the number of revolutions per second, 50 x diameter, 2-5) should =0-55.

Hence the speed =0-55 x 50=27-5 m. per second, or 99 km. per hour.

Again, Table V. proves, according to Commandant Dorand's experiments, that even at the present time it is possible to produce propellers giving the excellent efficiency of 84 per cent, under the most favourable running conditions, but only if the pitch ratio is greater than unity — that is, when the pitch is equal to or greater than the diameter.

It is further clear that, since the best reduced advance increases with the pitch ratio, the speed at which the machine should fly for the propeller (turning at a constant number of revolutions per minute) to give maximum efficiency is the higher the greater the pitch ratio. This is why propellers with a high pitch ratio, or the equivalent, a high maximum thrust, are more especially adaptable for high-speed aeroplanes. At the same time, they are equally efficient when fitted to slower machines, provided that the revolutions per minute are reduced by means of gearing.

These truths are only slowly gaining acceptance to-day — although the writer advocated them ardently long since, — and this notwithstanding the fact that the astonishing dynamic efficiency of the first motor-driven aeroplane which in 1903 enabled the Wrights, to their enduring glory, to make the first flight in history, was largely due to the use of propellers with a very high pitch ratio, that is, of high efficiency, excellently well adapted to the relatively low speed of the machine by the employment of a good gearing system.

The only thing that seemed to have been taught by this fine example was the use of large diameter propellers.

This soon became the fashion. But, instead of gearing

FLIGHT IN STILL AIR 63

down these large propellers, as the Wrights cleverly did, they were usually driven direct by the motor, and so that the latter could revolve at its normal number of revolutions the pitch had perforce to be reduced.

As the pitch decreased, so the maximum efficiency and the best reduced advance — that is, the most suitable flying speed — fell off, while at the same time the development of the monoplane actually led to a considerable increase in flying speed.

The result was that fast machines had to be equipped with propellers of very low efficiency which, even so, they were unable to attain, as the flying speed of the aeroplane was too high for them. At most these propellers might have done for a dirigible, but they would have been poor even at that.

Fortunately, a few constructors were aware of these facts, and to this alone we may ascribe the extraordinary superiority shown towards the end of 1910 by a few types of aeroplanes, among which we may name, without fear of being accused of bias, those of M. Breguet and the late M. Nieuport.

But, since then, progress has been on the right lines, and those who visited the last three Aero shows must have been struck with the general decrease in propeller diameter, which has been accompanied by an increase in efficiency and adaptability to the aeroplanes of to-day.

To take but one final example : the fast Paulhan-Tatin " Torpille," already referred to, had a pitch ratio greater than unity. For this reason its efficiency was estimated in the neighbourhood of 80 per cent.

The foregoing considerations may be summed up as follows : —

1. The same propeller gives an efficiency varying accord- ing to the conditions in which it is run, depending on its forward travel per revolution.

2. Each propeller has a speed of forward travel or advance enabling it to produce its highest efficiency.

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3. For propellers of identical type but different diameters the various speeds of forward travel corresponding to the same thrust are proportional to the diameters, whence arises the factor of reduced advance, which, in other words, is the ratio between the forward travel per revolution and the diameter.

4. The maximum efficiency of a propeller and its best reduced advance depend on its shape, and more especially on its pitch ratio.

Hitherto the propeller has been considered as a separate entity, but in practice it works in conjunction with a petrol motor, whether by direct drive or gearing.

But the engine and propeller together constitute the power-plant, and this new entity possesses, by reason of the peculiar nature of the petrol motor, certain properties which, differing materially from those of the propeller by itself, must therefore be considered separately.

First, we will deal with the case of a propeller driven direct off the engine.

Let us assume that on a truck forming part of a railway tram there has been installed a propelling plant (wholly insufficient to move the tram) consisting of a 50-h.p. motor running at 1200 revolutions per minute, and of a propeller, while a dynamometer enables the thrust to be constantly measured and a revolution indicator shows the revolutions per minute.

The tram being stationary, the motor is started.

The revolutions will then attain a certain number, 950 revolutions per minute for instance, at which the power developed by the motor is exactly absorbed by the propeller. The latter will exert a certain thrust upon the train (which, of course, remains stationary), indicated by the dynamo- meter and amounting to, say, 150 kg.

The power developed by the motor at 950 revolutions per minute is shown by the power curve of the motor, which we will assume to be that shown in fig. 12. This would give about 43 h.p. at 950 revolutions per minute.

FLIGHT IN STILL AIR 65

The useful power, on the other hand, is zero, since no movement has taken place.

Now let the train be started and run at, say, 10 km. per hour or 5 m. per second, the motor still continuing to run.

The revolutions per minute of the propeller would immediately increase, and finally amount to, say, 1010 revolutions per minute.

The power developed by the motor would therefore have increased and would now amount, according to fig. 12, to 45-5 h.p.

But at the same time the dynamometer would show a smaller thrust — about 130 kg.

But this thrust would, though in only a slight degree, have assisted to propel the train forward and the useful

power produced by the propeller would be =8*7 h.p.

The acceleration in rotary velocity and the decrease in thrust which are thus experienced are to be explained on the score that the blades, travelling forward at the same time that they revolve, meet the air at a smaller angle than when revolving while the propeller is stationary. In these conditions, therefore, the propeller turns at a greater number of revolutions, though the thrust falls off.

If the speed of the train were successively increased to 10, 15 and 20 m. per second, the following values would be established each time : —

The normal number of revolutions of the power-plant ;

The corresponding power developed by the motor ;

The useful power produced by the propeller.

We could then plot curves similar to that shown in fig. 15, giving for every speed of the train the corresponding motive power (shown in the upper curve) and the useful power (lower curve). The dotted lines and numbers give the number of revolutions.

The lower curve representing the variation in the useful power produced by the propeller according to the forward

66

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speed of travel is of capital importance, and will hereafter be referred to as the power-plant curve.

Usually the highest points of the two curves, L and M, do not correspond. This simply means that generally, and unless precautions have been taken to avoid this, the pro- peller gives its maximum thrust, and accordingly has its best reduced advance, at a forward speed which does not

950 40

20

k-d of fliffkt (in

Q i

FIG. 15.

enable the motor to turn at its normal number of revolu- tions, 1200 in the present case, and consequently to develop its full power of 50 h.p.

It is even now apparent, therefore, that one cannot mount any propeller on any motor, if direct-driven, and that there exists, apart altogether from the machine which they drive, a mutual relation between the two parts constituting the power-plant, which we will term the proper adaptation of the propeller to the motor.

FLIGHT IN STILL AIR 67

Its characteristic feature is that the highest points in the two curves representing the values of the motive power and the useful power at different speeds of flights lie in a perpendicular line (see fig. 16).

The highest thrust efficiency is then obtained from the propeller at such a speed that the motor can also develop its maximum power.

30

20

^

Spekd of fh$k

10 (5

FIG. 16.

25

The expression maximum power-plant efficiency will be used to denote the ratio of maximum useful power Mm (see fig. 15) developed at the maximum power LJ of which the motor is capable (50 h.p. in the case under consideration).

The maximum power-plant efficiency, it is clear, corre- sponds to a certain definite speed of flight Om. This may be termed the best speed suited to the power-plant.

If the adaptation of the propeller to the motor is good (as in the case of fig. 16), the maximum power-plant efficiency is the highest that can be obtained by mounting

68 FLIGHT WITHOUT FORMULA

direct-driven propellers belonging to one and the same family and of different diameters on the motor.

Hence there is only one propeller in any family or series of propellers which is well adapted to a given motor.

We already know that in a family of propellers the characteristic feature is a common value of the pitch ratio — supposing, naturally, that the propellers are identical in other respects. The conclusion set down above can there- fore also be expressed as follows : —

There can be only one propeller of given pitch ratio that is well adapted to a given motor. The diameter of the pro- peller depends on the pitch ratio, and vice versa.

Propellers well adapted to a given motor consequently form a single series such that each value of the diameter corresponds to a single value of the pitch, and vice versa.

According to the results of Commandant Dorand's experi- ments with the type of propellers which he employed, the series of propellers properly adapted to a 50 h.p. motor turning at 1200 revolutions per minute can be set out as in Table VI., which also gives the best speed suited to the power-plant in each case, and the maximum useful powers developed obtained by multiplying the power of the motor, 50 h.p., by the maximum efficiency as given in Table V.

To summarise :

1. The useful power developed by a given power-plant varies with the speed of the aeroplane on which it is mounted. The variation can be shown by a curve termed the char- acteristic power-plant curve.

2. To obtain from the motor its full power and from the propeller its maximum efficiency the propeller must be well adapted to the motor, and this altogether independently of the aeroplane on which they are mounted.

3. There is only a single series of propellers well adapted to a given motor.

4. For a power-plant to develop maximum efficiency the aeroplane must fly at a certain speed, known as the best speed suited to the power-plant under consideration.

FLIGHT IN STILL AIR

TABLE VI.

Pitch Ratio.	Propeller Diameters (in metres).	Propeller Pitch (in metres ; product of cols. 1 and 2).	Best Suitable Speed.	Maximum Power- plant Efficiency (from Table V.)	Maximum Useful Power developed (product of 50 h.p. and col. 6).
m.p.s.	km.p.h.
1	2	3 4	5	6	7
0-5	2-46	1-23	14-2	51	0-45	22-5
0-6	2-33	1-40	17-7	64	0-53	26-5
0-7	2-24	1-57	21-1	76	0-61	30-5
0-8	2-16	1-73	23-8	86	0-70	35
0-9	2'09	1-88	26-4	95	0-76	38
1-0	2-04	2-04	29-5	106	0-80	40
1-15	1-98	2-28 33-3	126	0-84	42

In conclusion, it will be advisable to remember that the conclusions reached above should not be deemed to apply with rigorous accuracy. Fortunately, practice is more elastic than theory. Thus we have already seen in the case of the angle of incidence of a plane that there is, round about the value of the best incidence, a certain margin within whose limits the incidence remains good. Just so we have to admit that a given power-plant may yield good results not only when the aeroplane is flying at a single best speed, but also when its speed does not vary too widely from this value.

In other words, a certain elasticity is acquired in applying in practice purely theoretical deductions, though it should not be forgotten that the latter indicate highly valuable principles which can only be ignored or thrust aside with the most serious results, as experience has proved only too well.

CHAPTER V

FLIGHT IN STILL AIR THE POWER-PLANT (concluded)

IN the last chapter we confined ourselves mainly to the working of the power-plant itself, and more particularly to the mutual relations between its parts, the motor and the propeller, without reference to the machine they are employed to propel. The present chapter, on the other hand, will be devoted to the adaptation of the power-plant to the aeroplane, and incidentally will lead to some con- sideration of the variable- speed aeroplane and of the greatest possible speed variation.

In Chapter II. particular stress was laid on the graph termed the essential curve of the aeroplane, which enables us to find the different values of the useful power required to sustain in flight a given aeroplane at different speeds, that is, at different angles of incidence and lift coefficients.

In fig. 17 the thin curve (reproduced from fig. 6, Chapter II.) is the essential aeroplane curve of a Breguet biplane weighing 600 kg., with an area of 30 sq. m. and a detrimental surface of 1-2 sq. m.

But in the last chapter particular attention was also drawn to the graph termed the power-pla-nt curve, which gives the values of the useful power developed by a given power-plant when the aeroplane it drives flies at different

In fig. 17 the thick curve is the power-plant curve, in the case of a motor of 50 h.p. turning at 1200 revolutions per

FLIGHT IN STILL AIR

71

minute and a propeller of the Chalais-Meudon type, direct- driven, well adapted to the motor, and with a pitch ratio of 0-7.

Table VI. (p. 69) gives the diameter and pitch of the propeller as 2-24 m. and 1-57 m. respectively. The maximum power-plant efficiency corresponds to a speed of

60

FIG. 17.

22-1 m. per second. The maximum useful power is 30-5 h.p. These are the factors which enable us to fix M, the highest point of the curve.

It will be clear that, by superposing in one diagram (as in fig. 17, which relates to the specific case stated above) the two curves representing in both cases a correlation between useful powers and speeds, and referring, in one case to the aeroplane, in the other to its power-plant, we should obtain

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some highly interesting information concerning the adapta- tion of the power-plant to the aeroplane.

The curves intersect in two points, Rj and R2, which means that there are two flight speeds, Oj. and O2, at which the useful power developed by the power-plant is exactly that required for the horizontal flight of the aeroplane. These two speeds both, therefore, fulfil the definition (see Chapter II.) of the normal flying speeds.

From this we deduce that a power-plant capable of sustaining an aeroplane in level flight can do so at two different normal flying speeds. But in practice the machine flies at the higher of these two speeds, for reasons which will be explained later.

These two normal flying speeds will, however, crop up again whenever the relation between the motive power and the speed of the aeroplane comes to be considered. Thus, when the motive power is zero, that is, when the aeroplane glides with its engine stopped, the machine can, as already explained, follow the same gliding path at two different speeds. The same, of course, applies to horizontal flight, since, as has been seen, this is really nothing else than an ordinary glide in which the angle of the flight-path has been raised by mechanical means, through utilising the power of the engine.

Let us assume that the ordinary horizontal flight of the aeroplane is indicated by the point Rl5 which constitutes its normal flight.

The speed ORj will be roughly 23 m. per second, and the useful power required, actually developed by the propeller, about 30 h.p.

According to Table II. (Chapter II.), the normal angle of incidence will be about 4°, corresponding to a lift coefficient of 0-038.

Let it be agreed that in flight, which is strictly normal, the pilot suddenly actuates his elevator so as to increase the angle of incidence to 6|° (lift coefficient 0-05), and hence necessarily alters the speed to 20 m. per second.

FLIGHT IN STILL AIR 73

From the thin curve in fig. 17 (and from Table II., on which it is based) it is clear that the useful power required to sustain the aeroplane at this speed will be 24 h.p.

On the other hand, according to the thick curve in the same figure, the power-plant at this same speed of 20 m. per second will develop a useful power of 30-3 h.p., giving a surplus of 6-3 h.p. over and above that necessary to sustain the machine. The latter will therefore climb, and climb at a vertical speed such that the raising of its weight absorbs exactly the surplus, NN' or 6-3 h.p., useful power

developed by the power-plant, that is, at a speed of

bOO

=about 0-79 m. per second.

Since this vertical speed must necessarily correspond to a horizontal speed of 20 m. per second, the angle of the climb, as a decimal fraction, will be the ratio of the two speeds, i.e.

0-79

=:0<0395=about 4 centimetres per metre=l in 25.

As a matter of fact, we have already seen that by using the elevator the pilot could make his machine climb or descend ; but by considering the curves of the aeroplane and of the power-plant at one and the same time, we gain a still clearer idea of the process.

Should the pilot increase the incidence to more than 6|° the speed would diminish still more, and fig. 17 shows that, in so doing, the surplus power, measured by the distance dividing the two curves along the perpendicular correspond- ing to the speed in question, would increase. And with it we note an increase both in the climbing speed and in the upward flight-path.

Yet is this increase limited, and the curves show that there is one definite speed, 01, at which the surplus of useful power exerted by the power-plant over and above that required for horizontal flight has a maximum value.

If, by still further increasing the angle of incidence, the speed were brought below the limit 01, the climb-

74 FLIGHT WITHOUT FORMULA

ing speed of the aeroplane would diminish instead of increasing.

Nevertheless, the upward climbing angle would still increase, but ever more feebly, until the speed attained another limit, Op, such that the ratio between the climbing speed to the flying speed, which measures the angle of the flight-path, attained a maximum.

Thus, there is a certain angle of incidence at which an aeroplane climbs as steeply as it is possible for it to climb.

If, when the machine was following this flight-path, the angle of incidence were still further increased by the use of the elevator, in order to climb still more, the angle of the flight-path would diminish. Relatively to its flight-path the aeroplane would actually come down, notwithstanding the fact that the elevator were set for climbing.

The same inversion of the effect usually produced from the use of the elevator would arise if the aeroplane were flying under the normal conditions represented by the point R2 in fig. 17. For a decrease in the angle of incidence through the use of the elevator would have the immediate and inevitable result of increasing the speed of flight, which would pass from O2 to Og, for instance. But this would produce an increase QQ' in the useful power developed by the power-plant over and above that required for horizontal flight, so that even though the elevator were set for descend- ing, the aeroplane would actually climb.

This inversion of the normal effect produced by the elevator has sometimes caused this second condition of flight to be termed unstable.

For if a pilot flying hi these conditions, and not aware of this peculiar effect, felt his machine ascending through some cause or other, he would work his elevator so as to come down. But the aeroplane would continue to ascend, gathering speed the while. The pilot, finding that his machine was still climbing, would set his elevator still further for descending until the speed exceeded the limit

FLIGHT IN STILL AIR 75

Op, and the elevator effect returned to its usual state and the machine actually started to descend. The pilot, unaware of the existence of this condition and brought to fly under it by certain circumstances (which, be it added, are purely hypothetical), would therefore regain normal flight by using his controls in the ordinary manner.

Nevertheless, one is scarcely justified in applying to this second condition of horizontal flight the term " unstable " — if employed in the sense ordinarily accepted in mechanics, — for one may well believe that a pilot, aware of its existence, could perfectly well accomplish flight under this condition by reversing the usual operation of his elevator.

Still, it would be a difficult proposition for machines normally flying at a low speed, since the speed of flight under the second condition (indicated by the point R2, fig. 17) would be lower still.

But in the case of fast machines the solution is obvious enough. For instance, according to Table II., the minimum speed of the aeroplane represented by the thin curve in fig. 17 is about 63 km. per hour, whereas in the early days of aviation the normal flying speed of aeroplanes was less.

Now, note that by making an aeroplane fly under the second condition the angle of the planes would be quite considerable. In the case in question the angle would be in the neighbourhood of 15°, which is about 10° in excess of the normal flying angle.

The whole aeroplane would therefore be inclined at an angle equivalent to some ten degrees to the horizontal, with the result that the detrimental surface (which cannot be supposed constant for such large angles) would be increased, and with it the useful power required for flight.

In practice, therefore, the power-plant would not enable the minimum speed Or2 to be attained, and the second condi- tion of flight would take place at a higher speed and at a smaller angle of incidence. Still, it would be practicable

76 FLIGHT WITHOUT FORMULA

by working the elevator in the reverse sense to the usual.*

Now let us just see how a pilot could make his aeroplane pass from normal flight to the second condition ; although, no doubt, in so doing we anticipate, for it is highly im- probable that any pilot hitherto has made such an attempt.

When the aeroplane is flying horizontally and normally, the pilot would simply have to set his elevator to climb, and continue this manoeuvre until the flight-path had attained its greatest possible angle. The aeroplane would then return (and very quickly too, if practice is in accordance with theory) to horizontal flight, and now, flying very slowly, it would have attained to the second condition of flight. At this stage it would be flying at a large angle to the flight- path, very cabre, almost like a kite.

The greater part of the useful power would be absorbed in overcoming the large resistance opposed to forward motion by the planes. It will now be readily seen that, under these conditions, any decrease in the angle of incidence would cause the machine to climb, since, while it would have but little effect on the lift of the planes, it would greatly reduce their drag.

By the process outlined above, the aeroplane would successively assume every one of the series of speeds between the two speeds corresponding to normal and the second condition of flight (i.e. it would gradually pass from Or1 to O2, fig. 17), though it would have to begin with climbing and descend afterwards.

But we know that the pilot has a means of attaining these intermediary speeds while continuing to fly horizontally, namely, by throttling down his engine. This, at all events, is what he should do until the speed of the machine had

* At present we are only dealing with the sustentation of the aeroplane. From the point of view of stability, which will be dealt with in subsequent chapters, it seems highly probable that the necessity of being able to fly at a small and at a large angle of incidence will lead to the employment of special constructional devices.

FLIGHT IN STILL AIR

77

reached a certain point 01 (fig. 18) corresponding to that degree of throttling at which the power-plant curve (much flatter now by reason of the throttling-down process) only continues to touch the aeroplane curve at a single point L. Below this speed, if the pilot continues to increase the angle of incidence by using the elevator, horizontal flight cannot be maintained except by quickly opening the throttle.

It would therefore seem feasible to pass from the normal to the second condition of flight, without rising or falling,

FIG. 18.

by the combined use of elevator and throttle. But up till now all this remains pure theory, for hitherto few pilots know how to vary their speed to any considerable extent, and probably not a single one has yet reduced this speed below the point 01 and ventured into the region of the second condition of flight, that wherein the elevator has to be operated in the inverse sense.

The reason for this view is that the aeroplane, when its speed approaches the point 01, is flying without any margin, and consequently is then bound to descend. If therefore it obeys the impulse of descending given by the elevator, it no longer responds to the climbing manipulation.

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As soon as the pilot perceives this,* he hastens to increase the speed of his machine again by reducing the angle of incidence and opening his throttle, whereas, in order to pass the critical point, he would in fact have to open the throttle but still continue to set his elevator to climb.

The possibility of achieving several different speeds by the combined use of elevator and throttle forms the solution to the problem of wide speed variation.

The greatest possible speed variation which any aeroplane is capable of attaining is measured by the difference between the normal and the second condition of flight. But, up to the present at any rate, the latter has not been reached, and the lowest speed of an aeroplane is that (indicated by 01, fig. 18) corresponding to flight at the " limit of capacity."

This particular speed, not to be mistaken for one of the two essential conditions of flight, is usually very close to that corresponding to the economical angle of incidence (see Chapter II.). Hence the economical speed constitutes the lower limit of variation, which has probably never yet been attained.

In the future, if the second condition of flight is achieved in practice, one will be able to fly at the lowest possible speed an aeroplane can attain. This conclusion may prove of considerable interest in the case of fast machines, for any reduction of speed, however slight, is then important.

The highest speed is that of the normal flight of an aero- plane. In the example represented in fig. 17 this speed is 23 m. per second, or about 83 km. per hour. Since the economical speed of the machine in question is about 66 km. per hour, the absolute speed variation would be 17 km. per hour, or, relatively, about 20 per cent. This, however, is a maximum, since the economical speed, as we know, is never attained in practice.

The above leads to the conclusion that the way to obtain

* He is the more prone to do this owing to the fact that, with present methods of design and construction, stability decreases as the angle of incidence is increased.

FLIGHT IN STILL AIR

79

a large speed variation is to increase the normal flying speed.

In the previous example we assumed that the 50 h.p. motor turning at 1200 revolutions per minute was equipped with a propeller with a 0-7 pitch ratio, well adapted, whose characteristic qualities are given in Table VI.

Now let us replace this propeller by another, equally well

rStc]

30

adapted, but with a pitch ratio of 1-15. According to Table VI. the diameter of this propeller would be 1-98 m. and its pitch 2-28 m. The best speed corresponding to the new propeller would be 33 m. per second, and the maximum useful power developed at this speed 42 h.p.

Now let the new power-plant curve (thick line) be super- posed on the previous aeroplane curve (see fig. 19). For the sake of comparison the previous power-plant curve is also reproduced in this diagram.

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The advantage of the step is clear at a glance. In fact, the normal flying speed increases from Orl — equivalent to 23 m. per second or 83 km. per hour — to Or\ — equivalent to 26 m. per second, or about 93 km. per hour. This in- creases the speed variation from 17 to 27 km. per hour, or from 20 to 29 per cent.

Again, the maximum surplus power developed by the power-plant over and above that required merely for sustentation, amounting to about 7 h.p. with the former propeller, now becomes about 12 h.p. The quickest climb-

7 x 75 ing speed therefore grows from - — =0-88 m. per second

uOO

12x75

to = 1'5 m. per second.

oOO

Hence, by simply changing the propeller, one obtains the double result of increasing the normal flying speed of the aeroplane together with its climbing powers. Nor is the fact surprising, but merely emphasises our contention that since highly efficient propellers can be constructed, it will be just as well to use them.

In order to gain an idea of the relative importance of increasing the pitch ratio when this ratio has already a certain value, we may superpose in a single diagram (fig. 20), on the aeroplane curve, all the power-plant curves representing the various propellers, well adapted, used with the same 50-h.p. motor turning at 1200 revolutions per minute, according to Table VI.

Firstly, it will be evident that a pitch ratio of 0-5 would not enable the aeroplane in question to maintain horizontal flight, since the two curves — that of the power-plant and of the aeroplane — do not meet. In fact, the pitch ratio must be between 0-5 and 0-6 — 0-54, to be exact — for the power- plant curve to touch the aeroplane curve at a single point. Horizontal flight would then be possible, but only at one speed and without a margin.

But as soon as the pitch ratio increases, the normal flying speed and the climbing speed increase very rapidly. On

FLIGHT IN STILL AIR

81

the other hand, once the pitch ratio amounts to 0-9, the advantage of increasing it still further, though this still exists, becomes negligible. Beyond 1-0 a further increase of pitch ratio (in the specific case in question) need not be considered. All of which are, of course, theoretical con- siderations, although they point to certain definite principles which cannot be ignored in practice — a fact of which

						;
I						7
1					.1
1 fc			4:	^	V £+- . — 0-8	\jti — -1-00 I	— — .
	$	^		7^-0-6 **0-S4^	"0-7 ^ Y> ,/
	yjr			^0-J
/			Speed	offtiyl	tfanffl.j.	erSecJ
1	1	0 / ]	J 2 ?IG. 20.	0 2	5 J	0 J	j

constructors, as already remarked, are now becoming cognisant.

At the same time, the reduction of the diameter necessi- tated by the use of propellers of great efficiency is not without its disadvantages, more especially in the case of monoplanes and tractor biplanes in which the propeller is situated in front. In these conditions, the propeller throws back on to the fuselage a column of air which be- comes the more considerable as the propeller diameter is

6

82

FLIGHT WITHOUT FORMULA

reduced, since practically only the portions of the blades near the tips produce effective work.

It is on this ground that we may account for the fact that reduction in propeller diameter has not yet, up to a point, given the good results which theory led one to expect.

But when the propeller is placed in rear of the machine

70

6C

50

4C

30

20

20

Sj teed oj flight

•Sec)

30

40

35 FIG. 21.

The figures at the side of the curve indicate the lift.

and the backward flowing air encounters no obstacle, there is every advantage in selecting a high pitch ratio, and we have already seen that M. Tatin, in consequence, on his Torpille fitted a propeller with a pitch exceeding the diameter.*

* It may also be noticed that the need for reducing the diameter gradually disappears as the power of the motor increases, because the diameter of propellers well adapted to a motor increases with the power of the latter.

FLIGHT IN STILL AIR

83

The use of propellers of high efficiency, therefore, obviously increases the speed variation obtainable with any particular aeroplane.

The lower limit of this speed variation has already been seen to be the economical speed of the aeroplane.

Now, it should be noted that, in designing high-speed machines, the use of planes of small camber and with a very heavy loading has the result of increasing the value of the economical speed. Thus, the Torpille, already referred to, appeared to be capable of attaining a speed of 160 km. per hour ; * but its economical speed would have been about 28 m. per second or 100 km. per hour. Fig. 21 shows, merely for the sake of comparison, the curve of an aeroplane of this type (weight, 450 kg. ; area, 12-50 sq. m. ; detrimental surface, 0-30 sq. m.) plotted from the following table.

TABLE VII.

+3 a	Speed Value.	fei	1 § £;?	C? c3 ^	1*	11!
1			gi 13 .3^	H *	** 't? .
1		1*1	" 5fi gn^, f° 8 X o	^•s	i|	ill
fs	m.p.s.	km.p.h.	"ill	ir	ill	&i	i*5
			Q £		3-x		p<2~
1	2	3	4	5	6	7	8
o-oio	60	216	0-0007	31kg.	87kg.	118 kg.	94 h.p.
0-020	42-4	158	0-0013	29	43	72	40
0-030	34-6	125	0-0020	31	29	60	28
0-040	30	108	0-0034	38	22	60	24
0-050	26-8	97	0-0055	49	18	67	24
0-058	24-9	90	0-0100	77	16	93	31

* If we allow it a detrimental surface of 0'30 sq. metre, which is certainly not enough.

84

FLIGHT WITHOUT FORMULA

The speed variation of such a machine would be 60 km. per hour =38 per cent.

If it could fly in the second condition of flight, i.e. at 90 km. per hour, the speed variation would be 70 km. per hour, or 44 per cent.

In a machine o£ similar type, able to attain a speed of 200 km. per hour (weight, 500 kg. ; area, 9 sq. m. ; detri-

					*f
I ^ 5				/	/
? •^ 3	CO "5 o		o «o ,	/
5	0 Ll	0 «• o ./	/
	V6_	Jx
			Speed c	fflykl	{in Tn/x.rS	y

30

35

45

50

40

FIG. 22.

The figures by the side of the curve indicate the lift.

mental surface, 0-03 sq. m.), whose characteristic curve is plotted in fig. 22, according to Table VIII., the economical speed would be 34 m. per second, or 125 km. per hour, giving a speed variation of 75 km. per hour, or 38 per cent. If it could attain the second condition of flight, i.e. 110 km. per hour, the variation would be 90 km. per hour, or 45 per cent.

Fortunately, as may be seen, the high-speed machine of

FLIGHT IN STILL AIR

85

the future should possess a high degree of speed variation. And in the case of really high speeds even the smallest advantage in this respect becomes of great importance. It may well be that the necessity for achieving the greatest

TABLE VIII.

"S	Speed Value.	hi	01 X '*-' C ~C5 di ^.	iil	|j	|||
56		s.-s g	PHjg 0<I<	'So ^	rp	fe3^
	I	j/J	g,ix|	ill	!»	III
	m.p.s.	km.p.h.	$J!	«l^		l|
			p c2				PS*-
1	2	3	4	5	6	7	8
o-oio	74-8	270	0-0007	35kg.	134 kg.	169 kg.	168 h.p.
0-020	53	190	0-0013	33	68	101	72
0-030	43-1	155	0-0020	35	45	80	46
0-040	37-4	135	0-0034	43	34	77	39
0-050	33-4	120	0-0055	55	27	82	37
0-058	31	111	o-oioo	86	24	110	46

possible speed variation will induce pilots of the extra high speed machines of the future to attempt, for alighting, to fly at the second condition of flight.* In this they will only imitate a bird, which, when about to alight, places its wings at a coarse angle and tilts up its body.

Fig. 20 further shows that when the pitch ratio is less than 0-8 the highest point of the power-plant curve lies to the left of the aeroplane curve. It only lies to the right of it when the pitch ratio is equal to or greater than 0-9. If the pitch ratio were 0-85, the highest point of the power- plant curve would just touch the aeroplane curve, and would hence correspond to normal flight.

* Attention is, however, drawn to the remarks at the bottom of p. 74.

86 . FLIGHT WITHOUT FORMULA

In Chapter IV. it was shown that the highest point of the power-plant curve corresponds — the propeller being supposedly well adapted to the motor — to a rotational velocity of 1200 revolutions per minute, the normal number of revolutions at which it develops full power. If, there- fore, this highest point lies to the left of the aeroplane curve, the motor is turning at over 1200 revolutions per minute when the aeroplane is flying at normal speed. On the other hand, if the highest point lies to the right of the aeroplane curve, in normal flight the motor will be running at under 1200 revolutions per minute.

In neither case will it develop full power. Moreover, there is danger in running the motor at too high a number of revolutions, particularly if it is of the rotary type. Only a propeller with a pitch ratio of 0-85 could enable the motor to develop its full power (in the special case in question).

This immediately suggests the expedient of keeping the motor running at 1200 revolutions per minute while allow- ing the propeller to turn at the speed productive of its maximum efficiency through some system of gearing. Thus we are brought by a logical chain of reasoning to the geared-down propeller, a solution adopted in very happy fashion in the first successful aeroplane — that of the brothers Wright.

Let us suppose that an aeroplane whose curve is shown by the thin line in fig. 23 has a power-plant curve repre- sented by the thick line in the same figure, the propeller direct-driven, having a pitch ratio of 1-15, and hence possess- ing (according to Commandant Dorand's experiments) 84 per cent, maximum efficiency.

Evidently, however good this power-plant might be when considered by itself, it would be very badly adapted to the aeroplane in question, since, firstly, it would only enable the machine to obtain the low speed Orx ; and, secondly, the maximum surplus of useful power, the measure of an aeroplane's climbing properties, would fall to a very

FLIGHT IN STILL AIR

87

low figure. Hence, the machine would only leave the ground with difficulty, and would fly without any margin. And all this simply and solely because the best speed, Om, suited to the power-plant would be too high for the aero- plane.

Now let the direct-driven propeller be replaced by another of the same type, but of larger diameter, and geared down in such fashion that the best speed suited to this power- plant corresponds to the normal flying speed O'15 of the aeroplane (see fig. 23).

FIG. 23.

The maximum useful power developed by this power- plant remains in theory the same as before, since the pro- peller, being of the same type, will still have a maximum efficiency of 84 per cent. The new power-plant curve will therefore be of the order shown by the dotted line in the figure.

It is clear that by gearing down we first of all obtain an increase of the normal flying speed, and secondly, a very large increase in the maximum surplus of useful power — that is, in the machine's climbing capacity. In practice, however, this is not a perfectly correct representation of

88 FLIGHT WITHOUT FORMULA

the case, since gearing down results in a direct loss of efficiency and an increase in weight. Whether or not to adopt gearing, therefore, remains a question to be decided on the particular merits of each case. Speaking very generally, it can be said that this device, which always introduces some complication, should be mainly adopted in relatively slow machines designed to carry a heavy load.

In the case of high-speed machines it seems better to drive the propeller direct, though even here it may yet prove desirable to introduce gearing.

This study of the power-plant may now be rounded off with a few remarks on static propeller tests, or bench tests. These consist hi measuring, with suitable apparatus, on the one hand, the thrust exerted by the propeller turning at a certain speed without forward motion, and, on the other, the power which has to be expended to obtain this result.

Experiment has shown that a propeller of given diameter, driven by a given expenditure of power, exerts the greatest static thrust if its pitch ratio is in the neighbourhood of 0-65.* On the other hand, we have seen that the highest thrust efficiency hi flight is obtained with propellers of a pitch ratio slightly greater than unity. Hence one should not conclude that a propeller would give a greater thrust hi flight simply from the fact that it does so on the bench. Thus, the propeller mounted on the Tatin Torpille, already referred to, which gave an excellent thrust hi flight, would probably have given a smaller thrust on the bench than a propeller with a smaller pitch.

Consequently, a bench test is by no means a reliable indication of the thrust produced by a propeller hi flight. Besides, it is usually made not only with the propeller alone but with the complete power-plant, in which case the result is even more unreliable owing to the fact that the power developed by an internal combustion engine varies with its speed of rotation.

For instance, suppose that a motor normally turning at * From Commandant Dorand's experiments.

FLIGHT IN STILL AIR 89

1200 revolutions per minute is fitted with a propeller of 1-15 pitch ratio which, when tested on the bench by itself, already develops a smaller thrust than a propeller of 0-65 pitch ratio ; the motor would then only turn at 900 revolu- tions per minute, whereas the propeller of 0-65 pitch ratio would let it turn at 1000 revolutions per minute, and hence give more power. The propeller with a high pitch ratio would therefore appear doubly inferior to the other, and this notwithstanding the fact that its thrust in flight would undoubtedly be greater.

A propeller exerting the highest thrust in a bench test must not for that reason be regarded as the best.

CHAPTER VI STABILITY IN STILL AIR

LONGITUDINAL STABILITY

AT the very outset of the first chapter it was laid down that the entire problem of aeroplane flight is not solved merely by obtaining from the " relative " air current which meets the wings, owing to their forward speed, sufficient lift to sustain the weight of the machine ; an aeroplane, in addition, must always encounter the relative air current in the same attitude, and must neither upset nor be thrown out of its path by a slight aerial disturbance. In other words, it is essential for an aeroplane to remain in equi- librium ; more, in stable equilibrium.*

We may now proceed to study the equilibrium of an aeroplane in still air and the stability of this equilibrium.

Since a knowledge of some of the main elementary principles of mechanics is essential to a proper understand- ing of the problems to be dealt with, these may be briefly outlined here.

* The very fact that an aeroplane remains in flight presupposes, as we have seen, a first order of equilibrium, which has been termed the equilibrium of sustentation, which jointly results from the weight of the machine, the reaction of the air, and the propeller-thrust. The mainten- ance of this state of equilibrium, which is the first duty of the pilot, causes an aeroplane to move forward on a uniform and direct course.

We are now dealing with a second order of equilibrium, that of the aeroplane on its flight-path. Both orders of equilibrium are, of course, closely interconnected, for if in flight the machine went on turning and rolling about in every way, its direction of flight could clearly not be maintained uniformly.

STABILITY IN STILL AIR

01

The most important of these is that relating to the centre of gravity.

If any body, such as an aeroplane, for instance (fig. 24), is suspended at any one point, and a perpendicular is drawn from the point of suspension, it will always pass, whatever the position of the body in question, through the same point G, termed the centre of gravity of the body.

The effect of gravity on any body, in other words, the

FIG. 24.

force termed the weight of the body, therefore always passes through its centre of gravity, whatever position the body may assume.

Another principle is also of the greatest importance in considering stability ; namely, the turning action of forces.

When a force of magnitude F (fig. 25), exerted in the direction XX, tends to make a body turn about a fixed point G, its action is the stronger the greater the distance, Gx, between the point G and the line XX. In other words, the turning action of a force relatively to a point is the greater the farther away the force is from the point.

Further, it will be readily understood that a force F', double the force F in magnitude but acting along a line YY separated from the fixed point G by a distance Gy,

92 FLIGHT WITHOUT FORMULA

which is just half of Gx, would have a turning force equal to F. In short, it is the well-known principle of the lever.

The product of the magnitude of a force by the length of its lever arm from a point or axis therefore measures the turning action of the force. In mechanics this turning action is usually known as the moment or the couple.

When, as in fig. 25, two turning forces are exerted in inverse direction about a single point or axis, and their

X/

FIG. 25.

turning moment or couple is equal, the forces are said to be in equilibrium about the point or axis in question.

For a number of forces to be in equilibrium about a point or axis, the sum of the moments or couples of those acting in one direction must be equal to the sum of the couples of those acting in the opposite direction.

It should be noted that in measuring the moment of a force, only its magnitude, its direction, and its lever arm are taken into account. The position of the point of its application is a matter of indifference. And with reason, for the point of application of a force cannot in any way influence the effect of the force ; if, for instance, an object

STABILITY IN STILL AIR 93

is pushed with a stick, it is immaterial which end of the stick is held in the hand, providing only that the force is exerted in the direction of the stick.

Before venturing upon the problem of aeroplane stability a fundamental principle, derived from the ordinary theory of mechanics, must be laid down.

FUNDAMENTAL PRINCIPLE. — So far as the equilibrium of an aeroplane and the stability of its equilibrium are con- cerned, the aeroplane may be considered as being suspended from its centre of gravity and as encountering the relative wind produced by its own velocity.

This principle is of the utmost importance and absolutely essential ; by ignoring it grave errors are bound to ensue, such, for instance, as the idea that an aeroplane behaves in flight as if it were in some fashion suspended from a certain vaguely-defined point termed the " centre of lift," usually considered as situated on the wings. An idea of this sort leads to the supposition that a great stabilising effect is produced by lowering the centre of gravity, which is thus likened to a kind of pendulum.

Now, it will be seen hereafter that in certain cases the lowering of the centre of gravity may, in fact, produce a stabilising effect, but this for a very different reason.

The " centre of lift " does not exist. Or, if it exists, it is coincident with the centre of gravity, which is the one and only centre of the aeroplane.

The three phases of stability, which is understood to comprise equilibrium, to be considered are :

Longitudinal stability.

Lateral stability.

Directional stability.

First comes longitudinal stability, which will be dealt with in this chapter and the next.

Every aeroplane has a plane of symmetry which remains vertical in normal flight. The centre of gravity lies in this plane. The axis drawn through the centre of gravity at right angles to the plane of symmetry may be termed

94 FLIGHT WITHOUT FORMULA

the pitching axis and the equilibrium of the aeroplane about its pitching axis is its longitudinal equilibrium.

Hereafter, and until stated otherwise, it will be assumed that the direction of the propeller-thrust passes through the centre of gravity of the machine. Consequently, neither the propeller-thrust nor the weight of the aeroplane, which, of course, also passes through the centre of gravity, can have any effect on longitudinal equilibrium, for, hi accordance with the fundamental principle set out above, the moments exerted by these two forces about the pitch- ing axis are zero.

Hence, in order that an aeroplane may remain in longi- tudinal equilibrium on its flight-path, that is, so that it may always meet the air at the same angle of incidence, all that is required is that the reaction of the air on the various parts of the aeroplane should be in equilibrium about its centre of gravity.

Now, in normal flight all the reactions of the air must be forces situated in the plane of symmetry of the machine. These forces may be compounded into a single resultant (see Chapter II.), which, for the existence of longitudinal equilibrium, must pass through the centre of gravity.

We may therefore state that : when an aeroplane is flying in equilibrium, the resultant of the reaction of the air on its various parts passes through the centre of gravity.

This resultant will be called the total pressure.

Let us take any aeroplane, maintained in a fixed position, such, for instance, that the chord of its main plane were at an angle of incidence of 10°, and let us assume that a hori- zontal air current meets it at a certain speed.

The air current will act upon the various parts of the aeroplane and the resultant of this action will be a total pressure of a direction shown by, say, P10 (fig. 26). Without moving the aeroplane let us now alter the direction of the air current (blowing from left to right) so that it meets the planes at an ever-decreasing angle, passing successively

STABILITY IN STILL AIR

95

from 10° to 8°, 6°, 4°, etc. In each case the total pressure will take the directions indicated respectively by P8, P6, P4, etc. Let G be the centre of gravity of the aeroplane.

Only one of the above resultants — P6, for instance — will pass through the centre of gravity. From this it may be deduced that equili- brium is only possible in flight when the main plane is at an angle of incidence of 6°.

Thus, a perfectly rigid unalterable aeroplane could only in practice fly at a single angle of in- cidence.

If the centre of gravity could be shifted by some means or other, to the posi- tion P4, for instance, the one angle of in- cidence at which the machine could fly would change to 4°. But this method for varying the angle of incidence has not hitherto been success- fully applied in practice.*

The same result, however, is obtained through an auxil- iary movable plane called the elevator.

It is obvious that by altering the position of one of the

* It will be seen hereafter that, if the method can be applied, it would have considerable advantages.

FIG. 2fi.

96

FLIGHT WITHOUT FORMULA

planes of the machine the sheaf of total pressures is altered. Thus, figs. 27 and 28 represent the total pressures in the case of one aeroplane after altering the position of the

FIG. 27.

elevator (the dotted outline indicating the main plane). If G is the centre of gravity, the normal angle of incidence passes from the original 4° to 2° by actuating the elevator.

STABILITY IN STILL AIR

97

Therefore, as stated in Chapter I., by means of the elevator the position of longitudinal equilibrium of an

FIG. 28.

aeroplane, and hence its incidence, can be varied at will.

The action of the elevator will be further considered in the next chapter.

But the longitudinal equilibrium of an aeroplane must

7

98

FLIGHT WITHOUT FORMULA

also be stable ; in other words, if it should accidentally lose

its position of equilibrium, the action of the forces arising

through the air current from the very fact of the change

in its position should cause it to regain this position instead of the reverse.

If we examine once again the sheaf of total pressures we may be able to gain an idea of how this condition of affairs can be brought about.

Returning again to fig. 26, let us suppose that by an oscillation about its pitching axis — the move- ment being counter-clock- wise — the angle of the planes, which is normally 6° since the total pressure P6 passes through the centre of gravity, decreases to 4°, the resultant of pressure on the aeroplane in its new position will have the direction P4 ; hence this resultant will have, relatively to the pitching axis, a moment acting clockwise, which

will therefore be a righting couple since it opposes the

oscillation which called it into being.

The same thing would come to pass if the oscillation was

in the opposite direction.

In this case, therefore, equilibrium is stable.

On the other hand, if the sheaf of pressures was arranged

as in fig. 29, the pressure P4 would exert an upsetting

FIG. 29.

STABILITY IN STILL AIR

99

En te.rt.rty Ectgre.

0-1

0-2

0-3

0-4

0-5

couple relatively to the pitching axis, and equilibrium would be unstable.

The stability or instability of longitudinal equilibrium therefore depends on the relative positions of the sheaf of total pressures and of the centre of gravity, and it may be laid down that when the line of normal pressure is in- tersected by those of the neighbouring total pres- sures at a point about the centre of gravity, equilibrium is stable, whereas it is unstable in the reverse case.

Several experimenters, and among them notably M. Eiffel, have sought to determine by means of tests with scale models the position of the total pressures corresponding to ordinary angles of incidence. Hitherto M. Eiffel's researches have been confined to tests on model wings and not on complete machines, .but the latter are now being employed. Moreover, the results do not indi- v on , . ,,,

FIG. 30. — Angles t of the chord and the wind.

cate the actual position

and distribution of the pressure itself, but only the point at which its effect is applied to the plane, this point being known as the centre of pressure.

The results of these tests have been plotted in two series of curves which give the position of the centre of pressure with a change in the angle of incidence. Figs. 30 and 31

0-6

0-7

0-8

0-9

10

Irt

-400-300-20°-IO° 0° 10° 20° 30

40

100

FLIGHT WITHOUT FORMULAE

reproduce, by way of indicating the system, the two series of curves relating to a Bleriot XI. wing.

It has already been remarked that the point from which a force is applied is of no importance ; accordingly, a centre

-30V

-is*

FIG. 31.

of pressure is of value only in so far as it enables the direc- tion of the pressures themselves to be traced.

By comparing the curve shown in fig. 31 with the polar curves already referred to in previous chapters, one obtains

STABILITY IN STILL AIR

101

a means of reproducing both the position and the magnitude, relatively to the wing itself, of the pressures it receives at varying

FIG. 32. — Sheaf of pressures on a flat plane.

Figs. 32, 33, and 34 show the sheaf of these pressures in the case, respectively, of : A flat plane.

A slightly cambered plane (e.g. Maurice Farman). A heavily cambered plane (Bleriot XL). These diagrams, be it repeated, relate only to the plane by itself and not to complete machines.

* A description of the method may be found in an article published by the author in La Technique Aeronautique (January 15, 1912).

102

FLIGHT WITHOUT FORMULAE

Comparison of these three diagrams brings out straight away a most important difference between the flat and the two cambered planes. That relating to the flat plane, in fact, is similar in its arrangement to that shown in

FIG. 33. — Sheaf of pressures on a Maurice Farman plane.

fig. 26, which served to illustrate a longitudinally stable aeroplane.

The diagrams relating to cambered planes, on the other hand, are analogous, so far as the usual flying angles are concerned, to fig. 29, which depicted the case of a longi- tudinally unstable aeroplane.

Thus we can state that, considered by itself, a flat plane is longitudinally stable, a cambered plane unstable (the

STABILITY IN STILL AIR

103

A -HHffA-

A [in /

B

FIG. 34. — Sheaf of pressures on a Bleriot XI. plane.

104 FLIGHT WITHOUT FORMULA

latter statement, however, as will subsequently be seen, is not always absolutely correct). On the other hand, every one knows nowadays that flat planes are very inefficient, producing little lift with great drag.

Hence the necessity for finding means to preserve the valuable lifting properties of the cambered plane while counteracting its inherent instability. The bird, inciden- tally, showed that it is possible to fly with cambered wings. And it was by adopting this example and improving upon it that the problem was solved, by providing the aeroplane with a tail.

An auxiliary plane, of small area but placed at a con- siderable distance from the centre of gravity of the aero- plane, and therefore possessing a big lever arm relatively to the centre of gravity, receives from the air, when in flight the aeroplane comes to oscillate in either direction, a pressure tending to restore it to its original attitude. Since this pressure is exerted at the end of a long lever arm, the couples, which are always righting couples, are of considerably greater magnitude than the upsetting couples arising from the inherent instability of the cambered type itself.

The adoption of this device has rendered it possible to utilise the great advantage possessed by cambered planes. Of course it is true that a machine with perfectly flat planes would be doubly stable, by virtue both of its main planes and of its tail, but to propel a machine of this type would mean an extravagant waste of power.

Provided the tail is properly designed, there is nothing to fear even with an inherently unstable plane, and the full lifting properties of the camber are nevertheless retained.

Subsequently it will be shown that the use of a tail entirely changes the nature of the sheaf of pressures, which, in an aeroplane provided with a tail, and even though its planes are cambered, assumes the stable form corresponding to a flat plane.

STABILITY IN STILL AIR 105

The aeroplane therefore really resolves itself into a main plane and a tail.*

Assuming, once and for all, that the propeller-thrust passes through the centre of gravity, the longitudinal equilibrium of an aeroplane about the centre of gravity can be represented diagrammatically by one of the three figs., 35, 36, and 37.

In fig. 35 the tail CD is normally subjected to no pressure and cuts the air with its forward edge. In this case, equi- librium exists if the pressure Q (in practice equal to the weight of the machine) on the main plane AB passes through the centre of gravity G.

In fig. 36 the tail CD is a lifting tail, that is, normally it meets the air at a positive angle and therefore is sub- jected to a pressure q directed upwards. For equilibrium to be possible in this case the pressure Q on the main plane AB must pass in front of the centre of gravity G of the aeroplane, so that its couple about the point G is equal to the opposite couple q of the tail.

The pressures Q and q must be inversely proportional to the length of their lever arms. When compounded they produce a resultant or total pressure equal to their sum (and to the weight of the aeroplane), which, as we know, would pass through the centre of gravity.

Lastly, in fig. 37 the tail CD is struck by the air on its top surface and receives a downward pressure q. To obtain equilibrium the pressure Q on the main plane AB must pass behind the centre of gravity G, the couples exerted about this point by the pressures Q and q being, as before, equal and opposite. Once again, the pressures Q and q must be inversely proportional to the length of their lever arms. If compounded they would produce a resultant total pressure equal to their difference (and to the weight

* In the case of a biplane both the planes will be considered as forming only a single plane, a proceeding which is quite permissible and could, if necessary, be easily justified.

106 FLIGHT WITHOUT FORMULA

G B CD

FIG. 35.

c D

FIG. 36.

P-Q-W /Q

B

V

*

FIG. 37.

STABILITY IN STILL AIR

107

of the aeroplane), which would again pass through the centre of gravity.

A fourth arrangement (fig. 38), and the first to be adopted in practice — since the 1903 Wright and the 1906 Santos- Dumont machines were of this type — is also possible. It has lately been made use of again hi machines of the " Canard " type (e.g. in the Voisin hydro-aeroplane), and consists in placing the tail, which must of course be a lifting tail, in front of the main plane. The conditions of equilibrium are the same as in fig. 36.

In an aeroplane, to whichever type it belongs, the term

FIG. 38.

longitudinal dihedral, or Fee, is usually applied to the angle formed between the chords of the main and tail planes.

Hitherto the relative positions of the main plane and the tail have been considered only from the point of view of equilibrium. We have now to consider the stability of this equilibrium. For this purpose we must return to the sheaf of pressures exerted, not on the main plane alone, but on the whole machine, that is, we have to consider the sheaf of total pressures.

This is shown in fig. 39,* which relates to a Bleriot XI.

* At the time when this treatise was first published, no experiments had been made to determine the actual sheaf of pressures as it exists in practice. The accompanying diagrams were drawn up on the basis of the composition of forces.

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Pf

FIG. 39. — Sheaf of total pressures on a complete Bleriot XI. monoplane.

wing provided with a tail plane of one-tenth the area of the main plane, making relatively to the main plane a longi-

STABILITY IN STILL AIR 109

tudinal Vee or dihedral of 6°, and placed at a distance behind the main plane equal to twice the chord of the latter.

Let it be assumed that the normal angle of incidence of the machine is 6°, which would be the case if its centre of gravity coincided with the pressure P6, at G15 for in- stance.

An idea of the longitudinal stability of the machine in these conditions may be guessed from calculating the couple caused by a small oscillation, such as 2°.

Since the normal incidence is 6°, the length of the pressure P6 is equivalent to the weight of the machine. By measuring with a rule the length of P4 and P8, it will be found to be equal respectively to P6xO-74 and to P6Xl-23. The values of P4 and P8 therefore are the pro- ducts of the weight of the aeroplane multiplied by 0-74 and 1-23 respectively.

Further, the lever arms of these pressures will, on measure- ment, be found to be respectively 0-043 and 0-025 times the chord of the main plane.

By multiplying and taking the mean of the results ob- tained, which only differ slightly, it will be found that an oscillation of 2° produces a couple equal to 0-031 times the weight of the aeroplane multiplied by its chord.

This couple produced by an oscillation of 2° can obviously be compared to the couple which would be produced by an oscillation of 2° imparted to the arm of a pendulum or balance of a weight equal to that of the aeroplane.

For these two couples to be equal, the pendulum arm must have a length of 0-88 of the chord, or, if the latter be 2m., for instance, the arm would have to measure 1-76 m. Hence, the longitudinal stability of the machine under consideration could be compared to that of an imaginary pendulum consisting of a weight equal to that of the aero- plane placed at the end of a 1-76 m. arm. It is evident that the measure of stability possessed by such a pendulum is really considerable.

110 FLIGHT WITHOUT FORMULA

Having laid down this method of calculating the longi- tudinal stability of an aeroplane, fig. 39 may once again be considered.

To begin with, it is evident that if the centre of gravity is lowered, though still remaining on the pressure line P6, the longitudinal stability of the machine will be increased since, the pressure lines being spaced further apart, then- lever arms will intersect. Therefore, under certain condi- tions, the lowering of the centre of gravity may increase longitudinal stability, though this has nothing whatsoever to do with a fictitious " centre of lift." Besides, in practice the centre of gravity can only be lowered to a very small extent, and the possible advantage derived therefrom is consequently slight, while, on the other hand, it entails disadvantages which will be dealt with hi the next chapter.

Finally, the use of certain plane sections robs the lower- ing of the centre of gravity of any advantages which it may otherwise possess, a point which will be referred to in detail hereafter.

Returning to fig. 39 — the normal angle of incidence being 6°, and the non-lifting tail forming this same angle with the chord of the main plane, the tail plane will normally be parallel with the wind (see fig. 35).

If the centre of gravity, instead of being at Gl5 were at G2, on the pressure line P8, the tail would become a lifting tail (see fig. 36), having a normal angle of incidence of 2°. Calculating as before, the length of the arm of the imaginary equivalent pendulum is found to be only 0-63 of the chord, or 1-26 m. if the chord measures 2 m.

The aeroplane is therefore less stable than in the previous example.

On the contrary, if the centre of gravity were situated at G3, corresponding to a normal incidence of 4°, so that the tail is struck by the wind on its top surface at an angle of 2° (in other words, is placed at a " negative " angle of 2°, see fig. 37), the equivalent pendulum would have to have

STABILITY IN STILL AIR 111

an arm 3-50 m. long,* or about twice as long as when the normal incidence is 6°.

From this one would at first sight be tempted to conclude that the longitudinal stability of an aeroplane is the greater the smaller its normal flying angle, or, in other words, the higher its speed ; but, although this may be true in certain cases, it is not so in others. Thus, if the alteration in the angle of incidence were obtained by shifting the centre of gravity, the conclusion would be true, since the sheaf of total pressures would remain unaltered.

But if the reduction of the angle is effected either by diminishing the longitudinal dihedral or, and this is really the same thing, by actuating the elevator, the conclusion no longer holds good, for the sheaf of total pressures does change, and in this case, as the following chapter will show, so far from increasing longitudinal stability, a reduction of the angle of incidence may diminish stability even to vanishing point.

It should further be noted that the arrangement shown diagrammatically in fig. 37, which consists hi disposing the tail plane so that it meets the wind with its top surface in normal flight, is productive of better longitudinal stability than the use of a lifting tail.f This conclusion will be found to be borne out by fig. 40, showing the pressures exerted on the main plane by itself.

By measuring the couples, it is clear that if the centre of gravity is situated at G1? for instance, the plane is unstable, as we already knew ; but if the centre of gravity were placed far enough forward relatively to the pressures, at G2, for instance, a variation in the angle may set up righting couples even with a cambered plane. The couple resulting from a variation of this kind is the difference between the

* Actually, the arm is longer if the oscillation is in the sense of a dive than in the case of stalling, which is quite in agreement with the con- clusions which will be set out later.

t It will be seen later that this arrangement also seems to be excellent from the point of view of the behaviour of a machine in winds.

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PlS

PIO

A

1

FIG. 40.

B

STABILITY IN STILL AIR

113

couples of the pressure, before and after the oscillation, about the centre of gravity.

Cambered planes in themselves may therefore be rendered stable by advancing the centre of gravity.

This is not difficult to understand ; as a plane is further removed from the centre of gravity it begins to behave

P. 5

FIG. 41. — Sheaf of total pressures on a Maurice Farman aeroplane.

more and more like the usual tail plane. In these conditions the stability of an aeroplane becomes very good indeed, since it is assisted by main and tail planes alike.

This explains why the tail-foremost arrangement (see fig. 38) can be stable, for in this arrangement the tail, situated in front, really performs the function of an " un- stabiliser," which is overcome by the inherent stability of

8

114 FLIGHT WITHOUT FORMULA

the main plane owing to the fact that the latter is situated far behind the centre of gravity.

Fig. 40 (which relates to the pressures on the main plane) further shows that if the centre of gravity is low enough, at G\, for instance, a Bleriot XI. wing would become stable from being inherently unstable. This is the reason for the stabilising influence of a low centre of gravity, which the examination of the sheaf of total pressures already revealed.

For the sake of comparison, fig. 41 is reproduced, showing the sheaf of total pressures belonging to an aeroplane of the type previously considered, but with a Maurice Farman plane instead of a Bleriot XI. section.

The pressure lines are almost parallel.

Lowering the centre of gravity in a machine of this type would produce no appreciable advantage.

It will be seen that the pressure lines draw ever closer together as the incidence increases, and become almost coincident near 90°. This shows that if, by some means or other, flight could be achieved at these high angles — which could only be done by gliding down on an almost vertical path, the machine remaining practically horizontal, which may be termed " parachute " flight, or, more colloquially, a " pancake " — longitudinal stability would be precarious in the extreme, and that the machine would soon upset, probably sliding down on its tail. Parachute flight and " pancake " descents would therefore appear out of the question, failing the invention of special devices.

CHAPTER VII STABILITY IN STILL AIR

LONGITUDINAL STABILITY (cone

IN the last chapter it was shown that the longitudinal stability of an aeroplane depends on the nature of the sheaf of total pressures exerted at various angles of incidence on the whole machine, and that stability could only exist if any variation of the incidence brought about a righting couple.

But this is not all, for the righting couple set up by an oscillation may not be strong enough to prevent the oscillation from gradually increasing, by a process similar to that of a pendulum, until it is sufficient to upset the aeroplane.

The whole question, indeed, is the relation between the effect of the tail and a mechanical factor, known as the moment of inertia, which measures in a way the sensitive- ness of the machine to a turning force or couple.

A few explanations in regard to this point may here be useful.

A body at rest cannot start to move of its own accord. A body in motion cannot itself modify its motion.

When a body at rest starts to move, or when the motion of a body is modified, an extraneous cause or force must have intervened.

Thus a body moving at a certain speed will continue to move in a straight line at this same speed unless some force intervenes to modify the speed or deflect the trajectory.

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The effect of a force on a body is smaller, the greater the inertia or the mass of the latter.

Similarly, if a body is turning round a fixed axis, it will continue to turn at the same speed unless a couple exerted about this axis comes to modify this speed.

This couple will have the smaller effect on the body, the more resistance the latter opposes to a turning action, that is, the more inertia of rotation it possesses. It is this inertia which is termed the moment of inertia of the body about its axis. The moment of inertia increases rapidly as the masses which constitute the body are spaced further apart, for, in calculating the moment of inertia, the dis- tances of the masses from the axis of rotation figure, not

in simple proportion, but as their square. An example will make this principle, which enters into every problem con- cerning the oscillations of an aeroplane, more clear.

At O, on the axis AB (fig. 42) of a turning handle a rod XX is placed, along which two equal masses MM can slide, their respective distances from the point O always remain- ing equal. Clearly, if the rod, balanced horizontally, were forced out of this position by a shock, the effect of this disturbing influence would be the smaller, the further the masses MM were situated from the point O, in other words, the greater the moment of inertia of the system.

If the rod were drawn back to a horizontal position by means of a spring it would begin to oscillate ; these oscilla- tions will be slower the further apart the masses ; but, on the other hand, they will die away more slowly, for the

STABILITY IN STILL AIR 117

system would persist longer in its motion the greater its moment of inertia.

These elementary principles of mechanics show that an aeroplane with a high moment of inertia about its pitching axis, that is, whose masses are spread over some distance longitudinally instead of being concentrated, will be more reluctant to oscillate, while its oscillations will be slow, thus giving the pilot time to correct them. On the other hand, they persist longer and have a tendency to increase if the tail plane is not sufficiently large.

This relation between the stabilising effect of the tail and the moment of inertia in the longitudinal sense has already been referred to at the beginning of this chapter. It may be termed the condition of oscillatory stability.

In practice most pilots prefer to fly sensitive machines responding to the slightest touch of the controls. Hence the majority of constructors aim at reducing the longi- tudinal moment of inertia by concentrating the masses.

It should be added that the lowering of the centre of gravity increases the moment of inertia of an aeroplane and hence tends to set up oscillation, one of the disadvan- tages of a low centre of gravity which was referred to in the last chapter.

By concentrating the masses the longitudinal oscillations of an aeroplane become quicker and, although not so easy to correct, present one great advantage arising from their greater rapidity.

For, apart from its double stabilising function, the tail damps out oscillations, forms as it were a brake in this respect, and the more effectively the quicker the oscillations. The reason for this is simple enough. Just as rain, though falling vertically, leaves an oblique trace on the windows of a railway-carriage, the trace being more oblique the quicker the speed of travel, so the relative wind caused by the speed of the aeroplane strikes the tail plane at a greater or smaller angle when the tail oscillates than when it does not, and this with all the greater effect the quicker

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the oscillation. It is a question of component speeds similar to that which will be considered when we come to deal with the effect of wind on an aeroplane.

The oscillation of the tail therefore sets up additional resistance, which has to be added to the righting couple due to the stability of the machine, as if the tail had to move through a viscous, sticky fluid, and this effect is the more intense the quicker the oscillation. It is a true brake effect.

In this respect the concentration of the masses possesses a real practical advantage.

According to the last chapter, an entirely rigid aeroplane, none of whose parts could be moved, could only fly at a single angle, that at which the reactions of the air on its various parts are in equilibrium about the centre of gravity. In order to enable flight to be made at varying angles the aeroplane must possess some movable part — a controlling surface.

Leaving aside for the moment the device of shifting the centre of gravity (never hitherto employed), the easiest method would be to vary the angle formed by the main plane and the tail, i.e. the longitudinal dihedral.

The method was first adopted by the brothers Wright, and is even at the present time employed in several machines. Very powerful in its effect, the variations in the angle of the tail plane affect the angle of incidence by more than then' own amount, and this hi greater measure the bigger the angle of incidence.

Figs. 43 and 44 represent two different positions of the sheaf of total pressures on an aeroplane with a Bleriot XI. plane, and a non-lifting tail of an area one-tenth that of the mam plane and situated in rear of it at a distance equal to twice the chord. In fig. 43 the tail plane forms an angle of 8° with the chord of the main plane ; in fig. 44 this angle is only 6°.

If the centre of gravity is situated at Gx, the normal angle of incidence passes from 4° in the first case to 2° in the second. This variation in the angle of incidence is

STABILITY IN STILL AIR

119

FIG. 43. — Sheaf of total pressures on a Bleriot XI. monoplane with a longitudinal V of 8°.

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PlO P8

PS

Po

ill

FIG. 44. — Sheaf of total pressures on a Bleriot XI. monoplane with a longitudinal V of 6°.

STABILITY IN STILL AIR 121

therefore integrally the same as that of the angle of the tail plane.

If the centre of gravity is at G2, the normal angle of incidence would pass from 6° to 3|°, and would therefore vary by 2|° for a variation in the angle of the tail of only 2°.

Lastly, if the centre of gravity is at G3, the normal angle of incidence would pass from 8° to 5°, a variation equal to one and a half times that of the angle of the tail.

A comparison of figs. 43 and 44 further shows that the lines of total pressure are spaced further apart the greater the longitudinal dihedral. Now, other things being equal, the farther apart the lines of pressure the greater the longi- tudinal stability of an aeroplane. Hence the value of the longitudinal dihedral is most important from the point of view of stability.

If the tail plane (non-lifting) is normally parallel to the relative wind, the longitudinal dihedral is equal to the normal angle of incidence. But if a lifting tail is employed, the longitudinal dihedral must necessarily be smaller than the angle of incidence (this is clearly shown in fig. 36). If the normal angle of incidence is small, as in the case of large biplanes and high-speed machines, the longitudinal dihedral is very small indeed and stability may reach a vanishing point.

But if, in normal flight, the tail plane meets the wind with its upper surface (i.e. flies at a negative angle), the longitudinal dihedral, however small the normal angle of incidence, will always be sufficient to maintain an excellent degree of stability. This conclusion may be compared with that put forward in the previous chapter in regard to the advantage of causing the tail to fly at a negative angle.

The foregoing shows that the reduction of the angle of incidence by means of a movable tail plane — i.e. by alter- ing the longitudinal dihedral — has the disadvantage that every alteration in the position of the tail plane brings

122 FLIGHT WITHOUT FORMULAE

about a variation in the condition of stability of the aeroplane.

By plotting the sheaf of total pressures corresponding to very small values of the longitudinal dihedral, it would soon be seen that if the latter is too small, equilibrium may become unstable.

A machine with a movable tail and normally possessing but little stability — such, for instance, as a machine whose tail lifts too much — may lose all stability if the angle of incidence is reduced for the purpose of returning to earth. This effect is particularly liable to ensue when, at the moment of starting a glide, the pilot reduces his incidence, as is the general custom.

Losing longitudinal stability, the machine tends to pursue a flight-path which, instead of remaining straight, curls downwards towards the ground, and at the same time the speed no longer remains uniform and is accelerated.

The glide becomes ever steeper. The machine dives, and frequently the efforts made by the pilot to right it by bringing the movable tail back into a stabilising position are ineffectual by reason of the fact that the tail becomes subject, at the constantly accelerating speed, to pressures which render the operation of the control more and more difficult.

In the author's opinion, the use of a movable tail is dangerous, since the whole longitudinal equilibrium depends on the working of a movable control surface which may be brought into a fatal position by an error of judgment, or even by too ample a movement on the part of the pilot.

For, apart from the case just dealt with, should the movable tail happen to take up that position in which the one angle of incidence making for stability is that corresponding to zero lift, i.e. when the main plane meets the wind along its " imaginary chord " (see Chapter I.), longitudinal equilibrium would disappear and the machine would dive headlong.

In this respect, therefore, the movements of a movable

STABILITY IN STILL AIR 123

tail should be limited so that it could never be made to assume the dangerous attitude corresponding to the rupture or instability of the equilibrium.

A better method is to have the tail plane fixed and rigid, and, hi order to obtain the variations in the angle of in- cidence required in practical flight, to make use of an auxiliary surface known as the elevator.

Take a simple example, that of the aeroplane diagram- matically shown in fig. 45, possessing a non-lifting tail

C D £

FIG. 45.

plane CD, normally meeting the wind edge-on, to which is added a small auxiliary plane DE, constituting the elevator, capable of turning about the axis D.

So long as this elevator remains, like the fixed tail, parallel to the flight-path, the equilibrium of the aeroplane will remain undisturbed. But if the elevator is made to assume the position DE (fig. 46), the relative wind strikes its upper surface and tends to depress it. Hence the incidence of the main plane will be increased until the couple of the pressure Q exerted about the centre of gravity, and the couple of the pressure q' exerted on the elevator, together become equal to the opposite moment of the pressure q on the fixed tail.

Again, if the elevator is made to assume the position

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DE2 (fig. 47), the incidence decreases until a fresh condition of equilibrium is re-established.

Each position of the elevator therefore corresponds to

FIG. 46.

one single angle of incidence ; hence the elevator can be used to alter the incidence according to the requirements of the moment.

FIG. 47.

It will be obvious that the effectiveness of an elevator depends on its dimensions relatively to those of the fixed tail, and, further, that if small enough it would be incapable, even in its most active position, to reduce the angle of

STABILITY IN STILL AIR 125

incidence to such an extent as to break the longitudinal equilibrium of the aeroplane.

This, in the author's opinion, is the only manner in which the elevator should be employed, for the danger of increasing the elevator relatively to the fixed tail to the point even of suppressing the latter altogether has already been referred to above.

In the position of longitudinal equilibrium corresponding to normal flight, the elevator, in a well-designed and well- tuned machine, should be neutral (see fig. 45). It follows that all the remarks already made with reference to the important effect on stability of the value of the longitudinal dihedral apply with equal force when the movable tail has been replaced by a fixed tail plane and an elevator.

The extent of the longitudinal dihedral depends on the design of the machine, and more especially on the position of the centre of gravity relatively to the planes, and on its normal angle of incidence, which, again, is governed by various factors, and in chief by the motive power.

The process of tuning-up, just referred to, consists prin- cipally in adjusting by means of experiment the position of the fixed tail so that normally the elevator remains neutral. Tuning-up is effected by the pilot ; in the end it amounts to a permanent alteration of the longitudinal dihedral ; where- fore attention must be drawn to the need for caution in effecting it.

There are certain pilots who prefer to maintain the longitudinal dihedral rather greater than actually necessary (i.e. with the arms of the V close together), with the con- sequence that their machines normally fly with the elevator slightly placed in the position for coming down, or meeting the wind with its upper surface. In the case of machines with tails lifting rather too much, the practice is one to be recommended, for machines of this description are dangerous even when possessing a fixed tail, since if the elevator is moved into the position for descent the longi- tudinal dihedral is still diminished, though in a lesser

126 FLIGHT WITHOUT FORMULA

degree, and if it were already very small, stability would disappear and a dive ensue.

Therefore the tuning-up process referred to has this advantage in the case of an aeroplane with a fixed tail exerting too much lift, that it reduces the amplitude of dangerous positions of the elevator and increases the amplitude of its righting positions.

If the size of the elevator is reduced, with the object of preventing loss of longitudinal equilibrium or stability, to such a pitch as to cause fear that it would no longer suffice to increase the angle of incidence to the degree required for climbing, an elevator can be designed which would act much more strongly for increasing the angle than for reducing it, by making it concave upwards if situated in the tail, or concave downwards if placed in front of the machine.

For it may be placed either behind or in front, and analogous diagrams to those given in figs. 46 and 47 would show that its effect is precisely the same in either case.

But it should also be noted that if an elevator normally possessing no angle of incidence is moved so as to produce a certain variation in the angle of incidence of the main plane, of 2°, for instance, the angle through which it must be moved will be smaller in the case of a front elevator than in that of a rear elevator, the difference between the two values of the elevator angle being double (i.e. 4° in the above case) that of the variation in the angle of incidence (assuming, of course, that front and rear elevators are of equal area and have the same lever arm).

This is easily accounted for by the fact that a variation in the angle of incidence, which inclines the whole machine, is added to the angular displacement of a front elevator, whereas it must be deducted from that of the rear elevator.

Thus, if we assume that the elevator must be placed at an angle of 10° to cause a variation in the incidence of 2°, the elevator need only be moved through 8° if placed in

STABILITY IN STILL AIR 127

front, whereas it would have to be moved through 12° if placed in rear.

A front elevator, therefore, is stronger in its action than a rear elevator. But it is also more violent, as it meets the wind first, which may tend to exaggerated manoeuvres. Finally, referring to the remarks in the previous chapters regarding the " tail-first " arrangement, the longitudinal stability of an aeroplane is diminished to a certain degree when the elevator is situated in front. These are no doubt the reasons that have led constructors to an ever-increasing extent to give up the front elevator.*

All these facts plainly go to show, as already stated, that stability does not necessarily increase with speed. Aero- planes subject to a sudden precipitate diving tendency only succumb to it when their incidence decreases to a large extent and their speed exceeds a certain limit, sometimes known as the critical speed, at which longitudinal stability, far from increasing, actually disappears altogether. The term critical speed is not, however, likely to survive long, if only because it refers to a fault of existing machines which, let us hope, will disappear in the future. And it would disappear all the more rapidly if the variations in the angle of incidence required in practical flight could be brought about, not by a movable plane turning about a horizontal axis, but by shifting the position of the centre of gravity relatively to the planes, which could be done by displacing heavy masses (such as the engine and passengers' seats, for example) on board or, also, by shifting the planes themselves.

In this case, as we have seen, the variations of the in- cidence would have no effect on the longitudinal dihedral, so that the sheaf of total pressures would not change, and then it would be true that stability increased with the speed. Then, also, there would be no critical speed.

* The placing of the propeller in front and the production of tractor machines — though, in the author's opinion, an unfortunate arrangement — has also formed a contributory cause.

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As stated previously, the horizontal flight of an aeroplane is a perpetual state of equilibrium maintained by con- stantly actuating the elevator. The idea of controlling this automatically is nearly as old as the aeroplane itself. But, as this question of automatic stability chiefly arises through the presence of aerial disturbances and gusts, its discussion will be reserved for the final chapter, which deals with the effects of wind on