Flight without formulae; simple discussions on the mechanics of the aeroplane

Front matter

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., and T. O'B. HUBBARD.

Chapter I Flight In Still Aie 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 horizontally, 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 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 their 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 angle,

* 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.

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 moving 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

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 expression " the drift of an aeroplane from its course in a side-wind," or 4 'drifting before a current."— TRANSLATOR.

FLIGHT WITHOUT FOKMUL.E

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

0.02 0.01 0-00

Drag.

FIG. 1. — Flat plane.

Drag.

FIG. 2. — Maurice Farraan plane.

The calculation of the lifting-power and the head-resistance produced by a given type of plane, moving through the air at a given angle of incidence and at a given speed, is exceedingly simple. To obtain the desired result all that

10? 10'

I

0.02

Drag.

FIG. 3. — Breguet plane.

0.00 o.oa P.OQ

O.Oi

0.00

002 Drag.

FIG. 4.— Bleriot XI. plane.

0.00 is needed is to multiply either the lift or the drag coefficients, corresponding to the particular angle of incidence, by the area of the plane (in square metres, or, if English measurements are adopted, in square feet) and by the square of the speed, in 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 ift Arpa Square of

Lltt Area< the Speed.

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

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 deductions regarding the speed of an aeroplane. The fore-and-aft equilibrium of an aeroplane, in 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.

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 loading 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. m.) 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 inflg. 3) to an angle of incidence of about 4°. What is its speed ?

The loading is -^- = 20 kg. per sq. m.

Square root of the loading = 4'47.

Inverse of the lift is ;r^-r = 25.

U°04<

Square root of inverse of the lift = 5.

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 hereafter, 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.

Lift.

Corresponding Angle of Incidence.

Inverse of Lift.

Square Root of Inverse of Lift.

Speed.

In m.p.s.

1 jbanp

PL, 3 g u -u

Ills

0-020

0° (about)

7-07

31-6

113-6

0-030

2° ,,

33*3

5-77

25-8

92'8

0-040

4°

5-00

22'3

80-3

0-050

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 wing 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 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 O'OOG. 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 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 therefore 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 0*01, 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 0*01 — 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 II

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 framework, 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 O01, and if, owing to a wind gust or to a manoeuvre corresponding to the sudden " flattening 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 coefficient 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,

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 ?

The loading = 9 =30 kg. per sq. m.

Square of the speed = 1300.

OA

Required lif t = T^ = 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 Ponnicr, measuring 8 sq. m., weighed approximately 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 considerable 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.

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 in 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-h.p. Deperdussin monocoque had a loading of 43*8 kg. per sq. m.

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

JO i

\L

Is

-— _

Lift 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 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 aeroplanes, attention may be drawn to a rule already laid down in a previous work,J which gives a rapid method of calculating 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. ?

Loading =—- =25 kg. per sq. m. ob

Square root of the loading = 5.

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

* In previous footnotes it has already been stated that the Deperdussin 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.

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

{ 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 in 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 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 aeroplane, the planes are not the only portions productive of drag, for they have to draw along the fuselage, or interplane 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 1J 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.

for a given detrimental surface by multiplying its area in square metres by the coefficient O08 (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. ^ SpeecL

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.