CHAPTER VIBALANCING THE AERODROME

◊[p045]CHAPTER VIBALANCING THE AERODROME

[p045]

By “balancing” I mean such an adjustment of the mean center of pressure of the supporting surfaces with reference to the center of gravity and to the line of thrust, that for a given speed the aerodrome will be in equilibrium, and will maintain steady horizontal flight. “Balance” and “equilibrium” as here used are nearly convertible terms.

Equilibrium may be considered with reference to lateral or longitudinal stability. The lateral part is approximately secured with comparative ease, by imitating Nature’s plan, and setting the wings at a diedral angle, which I have usually made 150°. Stability in this sense cannot be secured in what at first seems an obvious way—by putting a considerable weight in the central plane and far below the center of gravity of the aerodrome proper, for this introduces rolling. Thence ensues the necessity of carrying the center of gravity more nearly up to the center of pressure than would otherwise be necessary, and so far introducing conditions which tend to instability, but which seem to be imposed upon us by the circumstances of actual flight. With these brief considerations concerning lateral stability, I pass on to the far more difficult subject of longitudinal stability.

My most primitive observation with small gliding models was of the fact that greater stability was obtained with two pairs of wings, one behind the other, than with one pair (greater, that is, in the absence of any instinctive power of adjustment).

This is connected with the fact that the upward pressure of the air upon both pairs may be resolved into a single point which I will call the “center of pressure,” and which, in stable flight, should (apart from the disturbance by the propeller thrust) be over the center of gravity. The center of pressure in an advancing inclined plane in soaring flight is, as I have shown in “Aerodynamics,” and as is otherwise well known, always in advance of the center of figure, and moves forward as the angle of inclination of the sustaining surfaces diminishes, and, to a less extent, as horizontal flight increases in velocity. These facts furnish the elementary ideas necessary in discussing this problem of equilibrium, whose solution is of the most vital importance in successful flight.[p046]

The solution would be comparatively simple if the position of theCPcould be accurately known beforehand, but how difficult the solution is may be realized from a consideration of one of the facts just stated, namely, that the position of the center of pressure in horizontal flight shifts with the velocity of the flight itself, much as though in marine navigation the trim of a steamboat’s hull were to be completely altered at every change of speed. It may be remarked here that the center of pressure, from the symmetry of the aerodrome, necessarily lies in the vertical medial plane, but it may be considered with reference to its position either in the planeXY(cp1) or in the planeYZ(cp2). The latter center of pressure, as referred to in the planeYZ, is here approximately calculated on the assumption that it lies in the intersection of this vertical plane by a horizontal one passing through the wings half way from root to tip.

Experiments made in Washington, later than those given in “Aerodynamics,” show that the center of pressure, (cp1) on a plane at slight angles of inclination, may be at least as far forward as one-sixth the width from the front edge. From these later experiments it appears probable also that the center of pressure moves forward for an increased speed even when there has been no perceptible diminution of the angle of the plane with the horizon, but these considerations are of little value as applied to curved wings such as are here used. Some observations of a very general nature may, however, be made with regard to the position of the wings and tail.

In the case where there are two pairs of wings, one following the other, the rear pair is less efficient in an indefinite degree than the front, but the action of the wings is greatly modified by their position with reference to the propellers, and from so many other causes, that, as a result of a great deal of experiment, it seems almost impossible at this time to lay down any absolute rule with regard to the center of pressure of any pair of curved wings used in practice.

Later experiments conducted under my direction by Mr. E. C. Huffaker, some of which will appear in Part III, indicate that upon the curved surfaces I employed, the center of pressure moves forward with an increase in the (small) angle of elevation, and backward with a decrease, so that it may lie even behind the center of the surface. Since for some surfaces the center of pressure moves backward, and for others forward, it would seem that there might be some other surface for which it will be fixed. Such a surface in fact appears to exist in the wing of the soaring bird. These experiments have been chiefly with rigid surfaces, and though some have been made with elastic rear surfaces, these have not been carried far enough to give positive results.

The curved wings used on the aerodromes in late years have a rise of one in twelve, or in some cases of one in eighteen,24and for these latter the following empirical local rule has been adopted:[p047]

The center of pressure on each wing with a horizontal motion of 2000 feet per minute, is two-fifths of the distance from front to rear. Where there are two pairs of wings of equal size, one following the other, and placed at such a distance apart and with such a relation to the propellers as here used, the following wing is assumed to have two-thirds of the efficiency of the leader per unit of surface. If it is half the size of the leader, the efficiency is assumed to be one-half per unit of surface. If it is half as large again as the leader, its efficiency is assumed to be eight-tenths per unit of surface. For intermediate sizes of following wing, intermediate values of the efficiency may be assumed.

These rules are purely empirical and only approximate. As approximations, they are useful in giving a preliminary balance, but the exact position of the center of pressure is rarely determinable in either the horizontal or vertical plane, except by experiment in actual flight. The position of the center of gravity is found with all needed precision by suspending the aerodrome by a plumb-line in two positions, and noting the point of intersection of the traces of the line, and this method is so superior to that by calculation, that it will probably continue in use even for much larger constructions than the present.

The principal factor in the adjustment is the position of the wings with reference to the center of gravity, but the aerodrome is moved forward by the thrust of its propellers, and we must next recall the fact of experiment that as it is for constructional reasons difficult to bring the thrust line in the plane of the center of pressure of the wings, it is in practice sufficiently below them to tend to tip the front of the aerodrome upward, so that it may be that equilibrium will be attained only whenCP1isnotoverCG1.

In the discussion of the equilibrium, then, we must consider also the effect of thrust, and usually assume that this thrust-line is at some appreciable distance below the center of pressure.

We may conveniently consider two cases:

1. That the center of pressure is not directly over the center of gravity; that is,CG1−CP1=a, and estimate what the value of a should be in order that, during horizontal flight, the aerodrome itself shall be horizontal; or,[p048]

2. Consider that the center of pressure is directly over the center of gravity (CP1−CG1= 0), and in this case inquire what angle the aerodrome itself may take during horizontal flight.

First case. The diagram (Fig. 4) represents the resultants of the separate system of forces acting on the aerodrome, and these resultants will lie in a vertical medial plane from the symmetry of their disposition.

Letafrepresent the resultant of the vertical components of the pressure on the wings; the horizontal component will lie in the lineae.

FIG.4. Diagram showing relation under certain conditions of thrust,C. P.andC. G.

Let the center of gravity be in the linebd, and the resultant thrust of the propellers be represented bycd.

LetW= weight of aerodrome.

LetT= thrust of propellers.

Then if we neglect the horizontal hull resistance, which is small in comparison with the weight, equilibrium obtains whenW×ab=T×bd.

Second case. The diagram (Fig. 5) represents the same system of forces as Fig. 4, but in this case the point of support is directly over the center of gravityg, when the axis of the aerodrome is horizontal.

LetW= weight of aerodrome.

LetT= thrust of propellers.

LetR= distance ofCG2belowCP2=ag.

LetS= distance of thrust-line belowCP2=ad.

If now the aerodrome under the action of the propellers be supposed to turn about theCP2(or,a) through an angle α, so thatgtakes the positiong′, we[p049]obtain by the decomposition of the force of gravity an elementg′k=Wsin α which acts in a direction parallel to the thrust-line.

If we again neglect the horizontal hull resistance, equilibrium will be obtained when

kg′×ag′=T×ad′orWRsin α =TS∴ α = sin−1TSWR

kg′×ag′=T×ad′

orWRsin α =TS

WRsin α =TS

∴ α = sin−1TSWR

Fig. 5. Diagram showing relation under certain conditions of thrust,C. P.andC. G.

The practical application of these rules is greatly limited by the uncertainty that attaches to the actual position of the center of pressure, and this fact and also the numerical values involved may be illustrated by examples.

The weight was 12.5 kilos. On November 28, the steam pressure was less than 100 pounds, and the thrust may be taken at 4.5 kilos. The distancebdwas 25 cm.

Hence12.5 ×ab= 4.5 x 25 cm.ab= 9 cm.

Hence12.5 ×ab= 4.5 x 25 cm.

Hence

12.5 ×ab= 4.5 x 25 cm.

ab= 9 cm.

This appears to give the position ofCP1, butCP1is a resultant of the pressure on both wings, and its position is determined by the empirical rule just cited. We[p050]cannot tell in fact, then, with exactness how to adjust the wings so thatCG1−CP1may be 9 cm., and equilibrium was in fact obtained in flight when (the empirically determined)CG1−CP1= 3 cm.

Again, let it be supposed thatCP1was really overCG1. . . . The distance of the center of gravity below the center of pressure is 43 cm. =R.

Thenα = sin−14.5 × 2512.5 × 43= 12° nearly.

Then

α = sin−14.5 × 2512.5 × 43= 12° nearly.

The doubt as to the actual position of the resultant center of pressure, then, renders the application of the rule uncertain. In practice, we are compelled (unfortunately) after first calculating the balance, by such rules as the above, and after it has been thus found with approximate correctness, to try a preliminary flight. Having witnessed the actual conditions of flight, we must then readjust the position of the wings with reference to the center of gravity, arbitrarily, within the range which is necessary. This readjustment should be small.

FIG.6. Diagram showing effect of Pénaud tail.

In the preceding discussion it has been assumed that, if there is a flat tail or horizontal rudder, it supports no portion of the weight. This is not an indispensable condition but it is very convenient, and we shall assume it. In this case the action of the so-called Pénaud rudder becomes easily intelligible. This is a device, already referred to in Chapter II◊, made by Alphonse Pénaud for the automatic regulation of horizontal flight, and it is as beautiful as it is simple.

LetAB(Fig. 6) be a schematic representation of an aerodrome whose supporting surface isBb, and let it be inclined to the horizon at such an angle α that its course at a given speed may be horizontal. So far it does not appear that, if the aerodrome be disturbed from this horizontal course, there is any self-regulating power which could restore it to its original course; but now let there be added a flat tailACset at an angle −α with the wing. This tail serves simply for direction, and not for the support of the aerodrome, which, as already stated, is balanced so that theCGcomes under theCPof the wingBb.

It will be seen on a simple inspection that the tail under the given conditions is horizontal, and that, presenting its edge to the wind of advance, it offers no resistance to it, so that if the front rises and the angle α increases, the wind will strike on the under side of the tail and thereby tend to raise the rear and depress[p051]the front again. If the angle α diminish, so that the front drops, the wind will strike the upper surface of the tail, and equally restore the angle α to the amount which is requisite to give horizontal flight. If the angle α is not chosen originally with reference to the speed so as to give horizontal flight, the device will still tend to continue the flight in the straight line which the conditions impose, whether that be horizontal or not.

From this description of its action, it will be seen that the Pénaud tail has the disadvantage of giving an undulatory flight, if the tail is made rigid. This objection, however, can be easily overcome by giving to it a certain amount of elasticity. It does not appear that Pénaud gave much attention to this feature, but stress is laid upon it in the article “Flight,” in the ninth edition of the Encyclopædia Britannica, and I have introduced a simple device for securing it.

The complete success of the device implies a strictly uniform velocity and other conditions which cannot well he fulfilled in practice. Nevertheless, it is as efficient a contrivance for its object as has yet been obtained.

More elaborate devices have been proposed, and a number of them, depending for their efficiency upon the action of a variety of forces, have been constructed by the writer, one of which will be described later. This has the advantage that it tends to secure absolutely horizontal flight, but it is much inferior in simplicity to the Pénaud tail.

Apart from considerations about the thrust, theCPis in practice always almost directly over theCG, and this relationship is, according to what has been suggested, obtained by moving the supporting surfaces relatively to theCG, orvice versa, remembering, however, that, as these surfaces have weight, any movement of them alters theCGof the whole, so that successive readjustments may be needed. The adjustment is further complicated by another important consideration, namely, that those parts whichchangetheir weight during flight (like the water and the fuel) must be kept very near theCG. As the water and fuel tanks are fixed, it appears, then, that the center of gravity of the whole is practically fixed also, and this consideration makes the adjustment a much more difficult problem than it would be otherwise.25

PL. 11. STEEL FRAMES OF AERODROMESNOS. 4, 5, 6.1893, 1895 AND 1896◊

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