It will be seen from the foregoing statements of what has been accomplished with beating wings, that the principal questions are those of motive power and of proportion of surfaces to weight, and the reader will probably first inquire as to what is really the power developed by birds in their flight. The answer must unfortunately be, that it is not accurately known. A great many computations have been made, based upon more or less plausible assumptions, but none of these computations can be absolutely accepted as correctly based upon indisputably measured data.
This ceases to be surprising when we consider that there is no creature so willful, so swift, and so easily affrighted as the bird, and that once in the air, he will not lend himself to be measured experimentally. Mathematicians have, therefore, partly resorted to conjectures for their data. Thus Napier assumed that a swallow weighing 0.58 oz. must beat his wings 2,100 times per minute while going 33 1/2 miles per hour, in order to progress and sustain his weight, and that it therefore expended 1/13 of a horse power. In point of fact, the bird only beats about 360 times a minute, and is chiefly sustained by the vertical component of the air pressure on the under side of the wings and body, due to the speed, instead of by the direct blow of the wings downward, as supposed in the orthogonal theory already alluded to.
Other mathematicians, starting from the fact that a weight falls about 16 ft. during the first second, and in so dropping does work, have assumed that a bird in horizontal flight, being then sustained, performs a certain fraction of this work. It is evident, however, that if the bird does not drop, the fraction assumed is purely arbitrary, and that such calculations must be quite worthless.
Experiments to measure directly the power expended have proved failures, and resort has been had to indirect measurements.
Thus Dr. W. Smyth, of Edinburgh, succeeded in measuring with a dynamometer the strain exerted by a 12-oz. pigeon while flexing its wings, when excited by a current of electricity, and found it capable of raising 120 lbs. one foot high in a minute, or at the rate of 160 foot-pounds per pound of bird. Professor Marey performed the same experiment on the buzzard and on the pigeon, and ascertained the contractile strength of their muscles to be 18.46 and 19.91 lbs. to the square inch respectively;4 but as he was unable to measure satisfactorily the rapidity with which the muscles contracted, he did not calculate the footpounds.
Mr. Alexander, starting with the assumption that a 2-lb. pigeon makes 180 completed strokes per minute, each stroke with an amplitude of 1.5 ft. at the center of pressure, calculates the power exerted as being 2 X 180 X 1.5 = 540 foot-pounds per minute, or at the rate of 270 foot-pounds per pound of bird. This is plausible; but the most satisfactory computations are those made by Pénaud from observations of the direct velocity of ascent of various birds. From these he concludes that the pigeon, for instance, expends for rising 579 foot-pounds per minute, and that the proportion of horse power to weight is as follows:
This, however, is merely the work of elevation, such as would be performed upon a solid support, in addition to which the bird has to overcome the resistance of the air to his motion and to derive support from this mobile fluid. Pénaud calculates that this additional work amounts to over 1,000 foot-pounds per minute, so that the total work done by the pigeon in rising to a perch 35 ft. above the ground amounts to 1,650 foot-pounds per minute, or 1 horse power for every 20 lbs. Moreover, it must be remembered that the pectoral muscles of birds, which constitute their motor, comprise but one-quarter to one- sixth of their total weight, so that in this particular case the relative weight of the motor is only about 5 lbs. per horse power for the force exerted in rising.
These are formidable figures, but they cease to be discouraging when we reflect that the effort of rising is evidently a maximum, and that birds seldom perform it in a nearly vertical direction except for short distances, and that the exertion is clearly so severe that the feat is usually performed only by the smaller birds, which, as previously explained, must possess greater energy in proportion to their weight than those exceeding a few ounces. Heavy birds can only rise at angles less than 45°, and even then they exert for a short time far more than their mean strength, the latter being, for all animals, only a fraction of the maximum possible effort. Thus man, who is usually estimated as capable of exerting 0.13 horse power for 10 hours, can develop 0.55 horse power for 2 1/2 minutes, and nearly a full horse power for 3 or 4 seconds; and it seems probable that similar proportions obtain for birds, the emergency effort being three or four times the average performance, and the possible maximum about twice as great as the emergency effort.
Pénaud states that the ring-dove dispenses in full flight 217 foot-pounds per minute; but he does not give figures for this, so that they can be checked. Goupil estimates the work done by a pigeon weighing 0 .925 lbs. at 1,085 foot-pounds per minute in hovering and 119 footpounds per minute in flight; but the latter is arrived at by reasoning from analogy. It is evident that the power exerted in horizontal flight is much less than that required for rising or for hovering; but until a bird is taught to tow behind him some dynanometric arrangement at a regular rate of speed, and on a level course, it will be difficult to settle exactly what are the feet-pounds expended in ordinary performance.
In 1889 Captain de Labouret, an expert in the solution of balistic problems, analyzed mathematically two series of photographs of a gull weighing 1.37 lbs., and just starting out in flight with 5 wing beats per second, as obtained by by Professor Marey with the chrono- photographic process. The calculations showed that the bird expended in this act an average of 3,152 foot-pounds per minute. or 2 .303 foot-pounds per pound of his weight; and as Professor Marey shows that from his other observations of the reduced amplitude and rapidity of the wing beats, the same bird does only expend in full flight 6 of the effort required at starting, the conclusion may be drawn that the gull in full flight expends some 460 foot-pounds per minute for each pound of his weight.
This estimate seems plausible to me, and agrees with my own figures, but it is not accepted by all aviators. The Revue Scientifique of November 28, 1891, contains two articles disputing the conclusions -- one by Mr. V. Tatin, an expert aviator, who claims that the accelerations of the bird have been erroneously calculated; that the center of pressure under the wing is 4/9 of the distance from its root instead of 2/3, as usually assumed, and who figures out from the velocity of this new center of pressure, and from the known trajectory that the bird in full flight only expends from 33 to 197 foot-pounds per minute for each pound of his weight.
The second article is by Mr. C. Richet, the editor of the Revue Scientifique, who, having ascertained the volume of carbonic acid exhaled by a bird at rest, assumes, from experiments on other animals, that in full flight he will give out three times as much, and that the difference represents an effort of 105 foot-pounds per minute per pound of bird.
These two articles, being the most recent computations by earnest students of the subject, are here mentioned chiefly to illustrate how greatly aviators vary in estimates of the power expended, and how many elements have to be assumed in making such computations.
In the absence of direct measurements, and of positively satisfactory computation by others, of the feet-pounds expended in horizontal flight, I believe that an approximation may be obtained by analyzing and calculating the various elements which combine to make up the aggregate of the resistance to forward motion in horizontal progression and as this method promises to be useful in computing the power required by artificial flying machines, I venture to set it out at some length, applying it to the domestic pigeon as being more convenient to compare with the results of the calculations of others. For this purpose two dead pigeons were selected, weighing as near as practicable 1 lb. each. and their dimensions were accurately measured as follows:
|Piegon No. 1.||Pigeon No. 2.|
|Largest cross section of body||4.9 sq. in.||5.3 sq. in.|
|Largest cross section of edge of wings||5.02 sq. in.||4.88 sq. in.|
|Weight of bird, freshly killed||1 lb.||0.969 lb.|
|Horizontal area of both spread wings||90.35 sq. in.||99.86 sq. in.|
|Horizontal area of body projected||22.49 sq. in.||24.01 sq. in.|
|Horizontal area of tail spread||19.72 sq. in.||27.17 sq. in.|
|132.56 sq. in.||151.04 sq. in.|
These dimensions all require the application of coefficients in calculating their action upon the air. Thus the wings are concave, and give a greater sustaining power per square foot than a flat plane; the body is convex, and affords less than a plane, while the tail is slightly concave, but partly ineffective from its position. Previous experiments have indicated that, in the aggregate, the supporting power is about 30 per cent. more than that of a flat plane of equal area, so that in the calculations which follow the supporting surfaces will be assumed at 1.3 Sq. ft. to the pound instead of the 1 square foot to the pound which the average of the measurements seems to indicate.
It will be remembered that experiments with parachutes indicate a coefficient of resistance of 0.768 for the convex side and of 1.936 for the concave side, as compared with the plane of greatest cross section.
The cross sectional area of the body is assumed at 5 square inches or 0.03472 of a square foot. and to this a coefficient is applied of one-twentieth of a flat plane, or 0.05, in consequence of its elongated, fusiform shape. This agrees well with experiments on the hulls of ships of "fair" shape.
The cross sectional area of the wings is also taken at 5 square inches, or 0.03472 of a square foot; but the coefficient here assumed is about one-seventh, or 0.15, in consequence of its shape, which is ogival, or rather something like only half of a Gothic arch.
The friction of the air is omitted. as being entirely too small to affect the results in a case where so many coefficients have to be approximated.
The angle of flight is ascertained by selecting from the table previously given of air reactions, the coefficient which will give the nearest approximation to a sustaining "lift" to support the weight, and from this angle the "drift" is obtained to calculate the resistance of the surface.
The velocity Y is in feet per minute, and the pressure P on a plaue at right angles to the current by the Smeaton formula is in pounds per square foot. The following are the calculations:
20 miles per hour--V=1760 ft. P = 2 lbs.
Lift, 12°, 1.3 x 2 x 0.39 = 1.014 lbs. sustained.
|Drift, 12°||1.3 x 2 x 0.0828 = 0.21520 lb.||x 1760 = 378.7 ft. lbs.|
|Body resistance,||0.03472 x 2 x 0.05 = 0.00347 lb.||x 1760 = 6.1 ft. lbs.|
|Edge wings,||0.03472 x 2 x 0.15 = 0.01040 lb.||x 1760 = 18.3 ft. lbs.|
|0.22907 lb.||403.1 ft. lbs.|
30 miles per hour--V=2640 ft. P = 4.5 lbs.
Lift, 5°, 1.3 x 4.5 x 0.173 = 1.012 lbs. sustained.
|Drift, 5°||1.3 x 4.5 x 0.0152 = 0.08892 lb.||x 2640 = 234.7 ft. lbs.|
|Body resistance,||0.03472 x 4.5 x 0.05 = 0.00781 lb.||x 2640 = 20.6 ft. lbs.|
|Edge wings,||0.03472 x 4.5 x 0.15 = 0.02343 lb.||x 2640 = 61.9 ft. lbs.|
|0.12016 lb.||317.2 ft. lbs.|
40 miles per hour--V=3520 ft. P = 8 lbs.
Lift, 3°, 1.3 x 8 x 0.104 = 1.082 lbs. sustained.
|Drift, 3°||1.3 x 8 x 0.005343 = 0.05647 lb.||x 3520 = 198.7 ft. lbs.|
|Body resistance,||0.03472 x 8 x 0.05 = 0.01389 lb.||x 3520= 48.9 ft. lbs.|
|Edge wings,||0.03472 x 8 x 0.15 = 0.04166 lb.||x 3520= 146.6 ft. lbs.|
|0.11202 lb.||394.2 ft. lbs.|
50 miles per hour--V=4400 ft. P = 12.5 lbs.
Lift, 2°, 1.3 x 12.5 x 0.07 = 1.137 lbs. sustained.
|Drift, 2°||1.3 x 12.5 x 0.00244 = 0.03965 lb.||x 4400 = 174.5 ft. lbs.|
|Body resistance,||0.03472 x 12.5 x 0.05 = 0.02170 lb.||x 4400= 95.5 ft. lbs.|
|Edge wings,||0.03472 x 12.5 x 0.15 = 0.06510 lb.||x 4400= 286.5 ft. lbs.|
|0.12645 lb.||556.5 ft. lbs.|
60 miles per hour--V=5280 ft. P = 18 lbs.
Lift, 1 1/2°, 1.3 x 18 x 0.052 = 1.217 lbs. sustained.
|Drift, 1 1/2°||1.32 x 18 x 0.00136 = 0.0318 lb.||x 5280 = 167.9 ft. lbs.|
|Body resistance,||0.03472 x 18 x 0.05 = 0.0312 lb.||x 5280 = 164.7 ft. lbs.|
|Edge wings,||0.03472 x 18 x 0.15 = 0.0937 lb.||x 5280 = 494.7 ft. lbs.|
|0.1567 lb.||827.3 ft. lbs.|
These figures are probably somewhat in excess of the real facts in consequence of the adoption of slightly excessive coefficients for the resistance of the body and wing edges, which coefficients in full flight may be as much as one-third less than those which have been estimated.
It will be noticed that, as the velocity and the consequent air pressures increase, the angle of incidence required to obtain a sustaining reaction or "lift" diminishes, and so does, therefore, the "drift" or horizontal component of the normal pressure, while the "hull resistance," consisting of that of the body and edges of the wings, is at the same time increasing. There will therefore be some angle at which these various factors will so combine as to give a minimum of resistance, and this is probably for most birds at an angle of about 3°, which in the case of our calculated pigeon requires a speed of 40 miles per hour in order to sustain the weight.
This angle of minimum resistance depends upon the relative proportions of the bird -- i.e., upon the ratio between his surface in square feet per pound of weight, and the cross section of his body and wings, as well as their coefficient of resistance; and so, while the angle may not vary greatly, it needs to be ascertained for each case. Mr. Drzewiecki has calculated that for an aeroplane exposing a cross sectional area of 1 per cent. of its sustaining area (instead of the 7 per cent. which the measurements show for the pigeon), the angle of minimum resistance would be 1° 50' 45", and that it would be the same for all velocities. It does not follow, however, that the minimum of power required will coincide with the minimum of resistance, for the latter increases as the square, while the power grows as the cube of the speed. The calculations, therefore, show that the minimum of resistance occurs at 40 miles per hour, while the minimum of work done in foot-pounds is found at 30 miles per hour, and these two favorable speeds are about those observed from railway trains, as habitually practised by the domestic pigeon.
The estimates of the feet-pounds per minute indicate that the bird finds it less fatiguing to fly at 30 miles per hour than at 20; that his exertions are not much greater at 40 miles per hour, but that at 50 miles per hour he is expending rather more than his mean strength-the latter being probably about 425 foot-pounds per minute, nearly an average of the first four calculations, or about one-quarter of the maximum work done in rising, as estimated by Pénaud.
A flight of 60 miles within the hour is probably a severe exertion for the domestic pigeon, while the finer lines and greater endurance of the carrier pigeon enable him to maintain this speed for hours at a time; but there is reason to believe that this must be nearly the limit of his strength, and that homing birds who have made records of 70 and 75 miles per hour were materially aided by the wind.
The calculations therefore appear plausible, and to agree fairly well with the estimates arrived at with different methods by others. They indicate that if a flying machine can be built to be as efficient as the domestic pigeon its motor should develop one horse power for each 18 lbs. of its weight, provided it can give out momentarily about four times its normal energy, or that special devices, such as that of running down an incline or utilizing the wind, or some other contrivance are adopted to give it as tart and to enable it to rise upon the air.
The next question which the reader will probably want to ask, is as to the amount of supporting surfaces possessed by birds in proportion to their weight. Upon this point a good deal of information has been published; and in 1865 Mr. De Lucy greatly cheered aviators by publishing a paper in which he showed that the wing areas of flying animals diminish as the weight increases, from some 49 square feet to the pound in the gnat to 0.44 square feet to the pound in the Australian crane, and from which tables he inferred the broad law that the greater the weight and size of the volant animal. the less relative wing surface it required red.
As thus stated, the assertion is misleading. For inasmuch as the supporting surfaces will increase as the square, and the weight will grow as the cube of the homologous dimensions, It was to be expected that wing surfaces would not increase in the same ratio as the weight if the strength of the parts remained the same; and in 1869 Hartings published some tables of birds, in which he compared the square root of the wing surface with the cube root of the weight, and showed that their ratio became what he considered a somewhat irregular constant. Subsequent measurements and tables by Professor Marey have shown that this statement of Hartings is also slightly misleading, inasmuch as the socalled constant varies from 1.69 to 3.13, so that no broad law can be laid down as to any fixed relation between the surfaces and weight of birds of various sizes. The fact seems to be that while their structures are governed by the laws which limit the strength of materials (bones, muscles, feathers, etc.), yet there are differences in the resulting stresses, and in the consequent efficiency of the birds themselves, who are thereby led to adopt slightly different modes of flight; and in 1884 Müllenhoff published an able paper, in which he divided flying animals into six series, in accordance with the ratio between their weight and their wing surface, as well as their methods of flight. As the tables of De Lucy, Hartings Marey and Müllenhoff are all easily accessable in print they will not be repeated here; but the following table is considered more valuable than any of them. It has been compiled from "L'Empire de l'air" of Mr. Mouillard, a very remarkable book, published in 1881, which contains descriptions of the flight of many birds and accurate measurements of their surfaces and weights.
|Scientific Name.||Common Name.||Sq. Ft.|
speed for a
plane at 30
Miles per hr
|Galerita cristata I||Lark||3.18||0.315||24.6|
|Galerita cristala II||Lark||3.06||0.327||25.1|
|Passer domesticus I||Sparrow||2.42||0.414||28.2|
|Passer domesticus II||Sparrow||2.36||0.424||28.6|
|Larus rnelanocephalus I||Gull||2.35||0.426||28.6|
|Larus melanocephalus II||Gull||2.30||0.435||28.9|
|Turtur ægypticus||Egyptian Dove||2.27||0.441||29.2|
|Alcedo hispada I||Kingfisher||2.11||0.475||30.3|
|Alcedo hispada II||Kingfisher||2.11||0.475||30.3|
|Scolopax gallinula I||Snipe||1.96||0.510||31.4|
|Alcedo hispida III||Kingfisher||1.87||0.535||32.1|
|Scolopax gallinula II||Snipe||1.60||0.625||34.7|
|Ardea nycticorax||Night Heron||1.43||0.700||36.7|
|Columbia ægyptica I||Egyptian pigeon||1.37||0.730||37.5|
|Neophron percnopterus||Egyptian vulture||1.18||0.848||40.4|
|Columbia ægyptica II||Egyptian pigeon||1.13||0.885||41.3|
|Pelecanus anocrotales||Gray Pelican||0.732||1.365||51.3|
|Gyps fulvus||Tawny Vulture||0.679||1.473||53.3|
|Pterocles exustus||Running Pigeon||0.664||1.508||53.9|
|Procellaria gigantea||Giant Petrel||0.640||1.561||54.9|
|Anser sylvestris||Wild Goose||0.586||1.708||57.4|
|Anas clypeata, female||Duck||0.498||2.008||62.2|
|Anas clypeata, male||Duck||0.439||2.280||66.2|
Mr. Mouillard adopted a more rational method than other observers. Instead of merely measuring the surface of the wings, he laid the bird upon its back on a sheet of paper, projected the entire outline, and then measured the total area from which it gains support. The compilation has been made by Mr. Drzeweickt for a paper presented to the lnternational Aeronautical Congress at Paris in 1889, in which he states the general law more accurately than his predecessors, by calling attention to the fact that the ratio of weight to surface will vary somewhat with the structure of the bird, and that the result will be that those possessing the lesser proportionate surface must fly faster in order to obtain an adequate support at the same angle of incidence.
I have added the last column in the table, showing the speed required to sustain the weight of a flat plane loaded to the same proportion of weight to surface as the bird, at an angle of incidence of 3°. This speed merely approximates to the real flight of the bird, because it takes no account of the concavity of the wings, which, as previously explained, increases the effective bearing surface of the animal; but it would require experimenting with each and every bird tabulated in order to give the true and varying coefficients.
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