It has now come to be recognized that the design of aeroplanes can be based upon the information supplied by tests made with reduced models in an artificial current of air. Such tests determine the coefficients to be used in certain fundamental equations, and the problem of design is one of making use of the information available to attain a desired result. In many ways, aeroplane design and ship design are similar. In both we must estimate weight and strength from previous craft known to be satisfactory in operation. At the same time, for our particular design, a model must be tested in the model basin, if a ship, or in the wind tunnel, if an aeroplane, in order to know with certainty the power necessary to meet our limiting conditions.
As is well known, an aeroplane is sustained in the air by the reaction of the air upon its wings, which are propelled through the air at high speed inclined at a small angle. The total reaction or force of the air on an aeroplane wing is found to be well enough represented for the purposes of design by the formula:
R=KSV2. (1)
where K is a constant for any given wing shape and given angle of incidence.
S is the area in square feet of wing surface.
V is the velocity in feet per second relative to the air.
The vertical component of the total force of the air on the wings must balance the weight of the machine. This lift is represented by:
Lift=W=KySV2. (2)
Where W is the total weight of the aeroplane and Ky is a constant for given wing form and incidence.
The horizontal component of the total force of the air on the wings is in a direction opposed to the motion of the aeroplane and hence is a head resistance. It is represented by:
Rw=KxSV2. (3)
Where Rw is the head resistance of the wings and Kx is a constant for given wing form and incidence.
Unfortunately, it is not possible to make an aeroplane of wings alone, and a body or "fuselage" and tail, together with wing supports and landing gear or "chassis" is required. These parts do not add to the lift of the wings, but do add both weight and head resistance. The head resistance of the aeroplane fuselage, tail and chassis is best determined by testing a model in a wind tunnel after having removed its wings. This body resistance can be represented by:
Rb= KbV2. (4)
Where Kb is a constant. This formula is only used for an estimate, and it cannot be considered precise as it ignores the effect of change in inclination and ignores the experimental discovery that with long bodies where air friction is an important part of the resistance, the total resistance varies not as V2 but more nearly as V1.9. However, the difference is unimportant for the purpose of a preliminary design calculation.
The total resistance to motion is the quantity,
Rw+Rb=(KxS+Kb)V2, (5)
The effective power required to move the aeroplane with a speed V feet per second is therefore:
For wings and body of a given type, the above equations involve the following elements:
Rb= resistance of body, etc., without wings. There are thus but five variables, and from them the thrust and power may be calculated.
Kx. and Ky, are each some function of the incidence, but the relation is not a simple one, and it is best to use characteristic curves of Kx and Ky plotted on i from experimental model tests.
In the past, it has been the practice for an aeroplane builder to construct a machine and then to try various sizes and types of motor and propeller until a satisfactory performance was obtained. He then offered duplicates of this perfected machine for sale, making a guarantee of performance. There is no rational procedure in such a method, and besides being costly, as all trial and error methods are liable to be, is also contrary to sound engineering practice. It is much preferable to make the trials on the drawing board and in a preliminary design calculation, attempting to make the best use of the material available. After a design has been worked out which appears to justify full scale construction, it then may be further verified by model tests in a wind tunnel. Before risking money and time in building (and life in testing) a new design, its probable behavior can be predicted from its so-called characteristic curves. The characteristic curves as calculated are approximations and must be used with discretion, but they furnish valuable information upon which to decide whether the specifications are likely to be met with economy of material.
In Europe, aeroplanes are no longer bought by their trade names, but are required to meet definite specifications. The necessity for designing to meet given specifications has led to an intensive study of the aeroplane and very material improvement in safety, stability, radius of action and range of speed. After the inventor, has come the engineer and with him the testing laboratory and an analysis of a great part of a very foggy state of affairs.
Unfortunately, the engineer who is skilled in aeroplane design is such an asset to his firm that little or nothing is published regarding his detailed methods. For example, we know that in the past year a British biplane has been observed to leave the ground with a speed of about 40 miles per hour and, with fuel for three hours flight and a passenger, reach a speed in the air of over 90 miles per hour. Naturally, the details of this machine cannot be kept secret, as it is a commercial article. But it is not easy to learn how the designer went about his problem merely from an inspection of the aeroplane.
The great demand at the present time is safety, endurance and speed.
Safety is an indefinite term, but fairly well understood. It must depend on:
(1) Structural strength in the air.
(2) Controllability in the air.
(3) Inherent stability in the air.
(4) A wide margin of excess power always available.
(5) A high speed in the air.
(6) A low speed near the ground.
(7) An effective landing carriage.
(8) Protection against fire.
(9) A prudent pilot who keeps within prescribed limits of incidence and speed.
The problem of structural strength in the air is a matter of engineering. The forces involved are known and an ample factor of safety is introduced to take account of reversals and vibration. For example, the wings of the latest British Army machines are reported to be designed to support a load of from six to eight times the total weight of the machine.1 In all countries, structural failures are rare and strength no longer presents a problem of difficulty.
The matter of ease of control has been solved by the French, and it now remains to introduce insufficient inherent stability in the aeroplane without sacrificing too large a degree of its controllability. There is reason to believe that these two requirements are not incompatible.
1Engineering, London, May 22, 1914.
The theory of aeroplane stability has been placed on a firm mathematical basis, following the classical method of Rigid Dynamics, by Prof. G. H. Bryan of England.2
Mr. Leonard Bairstow3 of the National Physical Laboratory, England, has succeeded in determining the constants required in Bryan's equations of motion by direct experiments on small models. More recently still the Royal Air Craft Factory, England, has built a military biplane on conventional lines, but with a proportion of fin and tail area calculated by Bryan's method using Bairstow's model research. This machine has proved both easy to control and inherently stable in flight. It is reported that once clear of the ground, the pilot need only steer with his rudder even in gusty winds.4 A wide divergence from standard type in search of the same object is illustrated by the Dunne tailless biplane. The problem of stability cannot yet be dismissed as solved, for just now the theory is in advance of practice. It is only a question of a short time, however, when the stability of aeroplanes will be as generally understood as is the stability of ships; and, as in the case of ships, our problem is to get enough stability, but not too much. The technical discussion of aeroplane stability cannot well be developed in this paper.
Granted stability of the aeroplane, we have not secured safety unless the other considerations enumerated above are satisfied. The aeroplane being a dynamic machine, its presence aloft is absolutely dependent on its relative speed. With each speed there is one and only one attitude of the aeroplane (in the upright position), and it is here proposed to study the characteristics of the aeroplane from consideration of variations in speed and power. Speed and power are fundamental in design and fundamental for safe operation. Stability may well be considered apart during the first consideration of a design.
2"Stability in Aviation," Bryan, MacMillan.
3"The Stability of Aeroplanes," L. Bairstow, Flight, London, January 24, 1914.
4London Times, May 20, 1914, Col. Seely, Minister of War, discussing demonstration of RE-1 in a gusty wind of 28 miles per hour.
APPLICATION TO A PARTICULAR PROBLEM
Let it be a high speed military scout that is required for service with the marines on an expedition into the interior of some state. The nature of the service is such that quick action is required and obviously quicker scouting is necessary. Aside from racing machines which are not reliable, a speed of 90 miles per hour is as much as could be required. The marine column cannot afford to wait for its news, so that the scout would be required to proceed at full speed. Let us further decide that a flight of six hours, or 540 miles, is contemplated, and that a passenger must be carried, but no machine gun or bombs. The aeroplane is not to be armored. Safety in landing requires a speed of at least 45 miles an hour, and the aeroplane should get away from the ground at a speed not greater than 50 miles an hour. True, it would be preferable to leave the ground at 30 miles, but in view of the great speed required, it is unfair to specify so great a speed range in the present stage of the art.
The aeroplane is to be used in hostile country as a scout. It must operate at a height of 3000 feet when in danger and at other times must risk flying nearer the ground. A large climbing speed is, therefore, vital. A heavy biplane is usually good for 300 feet a minute. Let us here specify that it shall start to climb at 600 feet per minute.
We may now summarize the specification and consider the type of aeroplane:
(1) To carry two men.
(2) To leave ground at 50 miles an hour.
(3) To land at 45 miles an hour.
(4) To fly six hours at full power or 90 miles an hour.
(5) To climb 600 feet a minute.
It is evident that we shall have to deal with a very heavy machine. Two men for six hours at 90 miles an hour is considerably in advance of the standard machines. From a table of the weights and dimensions of the principal aeroplanes built in 1913-1914, it appears that there are some 10 machines which show a speed of about 45 to 80 miles an hour. The weights average about one ton (2240) pounds and the horse-power of their motors ranges between 90 and 160. Tank capacity is from four to five hours. All these machines are biplanes. Monoplane construction for machines weighing over one ton is unsuitable, although speeds over 100 miles an hour may be reached on small monoplanes. For our problem, there is a legitimate question whether a monoplane or a biplane type should be adapted. It will later be shown from considerations of aspect ratio and span that the biplane is preferable. We may take as a first approximation that our machine will weigh over one ton and will require about 120 horse-power to drive it. In view of the six-hour endurance requirement, it is important to use the smallest motor practicable to keep down the gasoline consumption per hour.
CHOICE OF MOTOR
We have two main types of motor to select between. The rotary air-cooled motor of low weight per horse-power, but rather uneconomical, and the stationary water-cooled type, heavier, but more economical of fuel. We must select that motor whose gross weight plux six hours fuel and oil supply is least.
Let t=duration of flight in hours.
V=speed in miles per hour, a constant.
D=distance flown in miles, a constant.
P=brake horse-power of motor, a constant.
a=gross pounds per horse-power of motor.
c=pounds oil and gas per horse-power hour.
The fuel and oil supply is therefore: tPc.
Tanks and pipes can be estimated at 1/6 of the weight of liquid provided. Then:
Total supply weight =7/6tPc.
Total motor weight =Pa.
Total power plant weight =P(a+7/6tc)=L.
For the best fixed type motor, the weight including radiator, water, magneto, etc., is about 5.3 pounds per horse-power, with a fuel and oil consumption of .66 pound per horse-power hour. This applies to the best German motors such as the "Benz" and "Mercedes." For rotary motors, such as the "Gnome," the unit weight is 3/5, and the fuel and oil consumption at least 3/2 the above respectively.5
Substituting these values in the above equation for L, the total power plant weights are:
120(5.3 + 7/6 x 6 x .66) = 120 x 9.92=1190 lbs. fixed type.
120(3/5 x 5.3 + 3/2 x 6 x 6.6) =120 x 10.10=1213 lbs. rotary type.
5Dr. Bendemann of the Adlershof Testing Laboratory in "Zeitschrift der Flugtechnik und Motorluftschiffahrt," April 25. 1914.
For six hours run there is then small choice between the two types so far as weight is concerned. However, in view of the greater reliability and robustness of the water-cooled type it should be selected. For any endurance longer than six hours at full power, the water-cooled type is obviously lighter.
Let us then search the published reports on motor tests for a suitable water-cooled motor of about 120 horse-power. The Salmson (Canton Unné) rated 120 horse-power motor, nine cylinders in star, develops 120-125 horse-power at 1300 revolutions per minute. It was run 100 hours with four stops of eight minutes each at the Chalais Meudon competition of the "Aero Club de France." It is, therefore, sufficiently reliable for our purposes. The weight is given as follows:
Pounds
Motor complete with magneto, 2 carburetors, oil pump, water pump,
piping, etc. 436
Double radiator, empty 53
Cooling water in radiator and motor 42
Total 531
Hourly consumption:
Gasoline 76.0
Oil 4.4
Total 80.4
Oil and fuel for six hours = 6 X 80.4 = 482.4
Tanks and pipes= 1/6 X 482.4 80.4,
Total 562.8
Power Plant Weight.—Grand Total 1094
This compares with a weight of 1190 pounds by taking Dr. Bendemann's figures 'quoted above for the best German motors. The check is reasonable and we are assured of having no handicap at the start from excessive motor weight.
Personnel.—The weight of personnel may be taken at 320 pounds, allowing for men of average weight.
Useful Weight.—The total useful weight, personnel and fuel is then about 883 pounds. In the 10 biplanes used above for comparison, the useful weight is from 30 to 50 per cent of the total. We will probably wish to provide a very substantial structure and may select the figure 35 per cent to be on the safe side. This indicates a total weight of about 2500 pounds which seems reasonable.
Fuselage.—It is next necessary to rough out the design of the body or fuselage. It must be of size sufficient to house the pilot and passenger, tanks, etc., and provide motor foundations. Mounting the propeller on the nose of the machine is most suitable for the Salmson motor. The diameter of the body will then be in the neighborhood of three feet at its maximum section. For a 90-mile machine, every effort must be made to keep down body resistance. A length of at least six diameters gives a good torpedo-shaped form. We then have a length of some 20 feet. This shell will be made up of ash or steel longitudinals with wooden hoops spaced at intervals. Its weight can best be estimated after plans are made. For our problem, we cannot expect to get sufficient stiffness on less than 200 pounds.
Chassis.—The chassis, or landing carriage, must also be designed if its weight is to be estimated with precision. Since our machine weighs over a ton and has a landing speed of between 45 and 50 miles an hour, we will need the most rugged chassis we can afford to carry. For military work, where landings on bad ground are likely, a four-wheeled chassis of Voisin or Bristol type is suitable. Effective springs and shock absorbers must be provided. Steel tubing is heavy but not easily splintered. At the same time, it is more difficult to repair in the field than wood. In case we use either material it is impossible to have the chassis too strong. Rather, the problem is to design as strong a chassis as possible for the weight allowed. Considering other heavy machines, we may be safe in allowing 200 pounds for the chassis.
Tails and Controls.—Our machine weighing over a ton must have wings of about 400 square feet area if we allow six pounds per square foot loading. This is the upper safe limit for such a great spread of wings. It will then be necessary to design the tail surfaces on the basis of wind tunnel tests to obtain inherent stability.
The horizontal tail surface in machines with good longitudinal stability is about one-fifth the wing area. The horizontal rudder area is included in this. Vertical fins and rudders may total one-tenth of the wing area. For purpose of weight estimate, we have then some 120 square feet of directive surface which may be taken to weigh complete with bracing some 100 pounds.
We may now assemble these rough weight estimates as follows:
Pounds
Gross motor weight 531
Oil, fuel and tanks 563
Propeller and hub 24
Personnel and equipment 320
Fuselage 200
Chassis 200
Directive surfaces 100
Biplane struts and wires 45
Total weight without wings 1983
Weight of Wings.—The weight of wings varies with the type of construction and factor of safety allowed. The Royal Air Craft Factory, England, finds that the following empirical formula well expresses the weight of their wings, including fabric and enamel:
Wing weigh t= 17A½ +0.16A.
Where A is the area in square feet.
We can then prepare the following table:
Area of Wings Weight of Wings Total Weight of Aeroplane
0 0 1983
100 186 2169
200 272 2255
300 343 2325
400 404 2387
The size of wings to use is a matter of judgment. Weight is saved by keeping the wings small, but practical considerations impose a limit. It is not safe to impose too large a load per square foot with wings of great area. Large slow biplanes are given a wing loading of four pounds while small fast monoplanes of the racing machine order show a wing loading of over 11 pounds per square foot. Values of wing loading tend to increase in the past year. The following curve represents current practice.
It appears that the curve drawn represents an upper limit, or for a 400 square foot wing six pounds is high and for a 300 square foot spread seven pounds is high.
On Fig. 2, are shown the curves for wing loadings of six, seven and eight pounds intersecting the curve of total weight of aeroplane. Thus, if we select a loading of six pounds a wing area of 400 square feet is needed, and for a loading of seven pounds a wing area of 353 square feet. A loading of eight pounds on an area 290 square feet by a reference to Fig. 1 is beyond current practice and might lead to structural difficulties.
Selection of Wing Form.—The aerodynamic laboratories have supplied the designer with the characteristics of a great number of wing forms. It now remains to select that one which is most suitable to our case. It appears to be easier to work backwards, and first determine what must be the properties of a wing to permit the speed range we require.
It was stated above that for any wing form, the force per unit area exerted by a wind of one foot per second velocity is a function of the angle of incidence of the wind. For simplicity, the angle of incidence for a cambered surface is defined as the angle between the chord and the wind. The unit force K has a horizontal component Kx. and a vertical component or lift Kyfor each angle of incidence.
To satisfy the conditions of our problem, we may solve equation 2 for Ky„ thus:
Similarly, for Kx we may solve equation 8. If we denote by e the efficiency of the propeller and by P the effective horse-power required to drive the aeroplane at full speed, the brake horsepower of the motor B is given by:
But by equation 8:
Substituting for P and solving for Kxwe have:
For a wing loading of seven pounds per square foot the surface S is 353 square feet. B=125 horse-power. The propeller efficiency at full speed may be taken at 78 per cent and no allowance made for loss of efficiency at slow speeds until the final calculation. The solution of equations 9 and 11 may be tabulated as below :
Similarly, for a loading of six pounds per square foot the wing area is 400 square feet with other constants unchanged. The following table may be prepared:
In order to support the weight of the aeroplane, any wing we choose must develop a lift coefficient Ky at each speed exactly equal to the Ky calculated as required for the given loading. Similarly, to drive the aeroplane at each speed the resistance coefficient Kxof the wing must be not greater than the Kx above calculated for the given loading. If Kxfor the wing at any speed is smaller than K, required, an advantage is gained, for the motor may then be run at reduced power.
The corresponding values of Kx and Ky required are plotted as two curves on Fig. 3, one for the six pound loading and one for the seven pound loading. The corresponding speeds are marked along the curves.
It is usual in the reports of aerodynamic experiments on wing forms to give a table of Kx and Ky for a series of angles of the chord to the wind. On Fig. 3 are then plotted the characteristic Kx Ky curves for a number of wing forms. These values have been taken from various sources and transposed to English units; i. e., pounds force per square foot for a wind velocity of one foot per second; normal barometer and temperature. The angles of incidence as above defined are marked along each wing curve.
The lower intersection of the characteristic curve for any wing with one of the curves of required Kx and Ky indicates the conditions under which that wing will be suitable. From the numbers marked along the curves, one can read off the speed of the aeroplane and the angle of incidence of the wings. For example, for a loading of seven pounds and wing marked "Bleriot," a speed of some 86 miles only can be reached with an incidence of about 2.5°. The upper intersection of the same curves shows a speed of a little over 50 miles for an incidence of about 19°. Obviously, the Bleriot wing cannot meet our specifications. A similar analysis can be made for each of the other wing curves taken in connection with each wing loading curve.
It is seen that with the seven pound loading several wings give a speed of 90 miles an hour, but there is difficulty in reaching the low speed below 50 miles. It should be remarked here that, in general, the wing curves intersect the loading curves in two points. However, in many cases the upper end of the wing curve cannot be produced far enough to obtain an intersection because the angle of maximum Ky, is reached for a small value of Kx. This, however, is in our favor as showing a Kx coefficient less than the maximum allowable on the loading curve for 125 horse-power. The upper intersection can then be found by drawing a horizontal line from the highest point of the wing characteristic curve until it cuts the two loading curves.
From a consideration of the diagram of Fig. 3, we would conclude to use a wing loading of six pounds per square foot. It is also noticed that all wings give a speed under 50 miles an hour, and three give speeds over 90 Miles an hour. The selection of one of these three wings is a matter of judgment.
Before the days when inherent longitudinal stability could be given an aeroplane, the designer was careful to use a wing with an incidence of at least two degrees on the theory that if the machine flies at a small angle, a sudden pitch may be enough to change that angle to negative and thus throw the pressure on the top of the wings. A good many men have been pitched out of their seats under such circumstances. It is now recognized, however, that the angle of incidence is an arbitrary quantity as defined. The chord in many wings may be inclined at a negative angle from 3° to 5° before the lift becomes zero. For example, wing Eiffel No. 37 lifts more at —3° than NPL No.16 at +1°. The inference is that they are equally far from the incidence of zero lift. Further experiments are required on this point and before completing a design the wing chosen should have its characteristic determined for all probable angles of incidence.
With a machine with a large non-lifting tail slightly inclined upwards, the tail in normal flight can be made to take a downward reaction of some 4 per cent of the total weight of the machine. In such a case, the center of gravity is a few inches in front of the center of pressure of the wings, and there will be one and only one angle of incidence at which these three forces balance. Any pitching of the aeroplane is the cause of a restoring couple to return the machine to its original trim.
The man must be strapped in always with a quick release toggle and flying at a good elevation when at full speed.
In view of adequate longitudinal stability7 it does not appear wise to shun the small angle with its consequent advantages of high speed on low power. The arguments that showed a small incidence to be dangerous for an unstable machine no longer have force.
It is therefore possible to choose for our design wing Eiffel No. 378 giving a speed of over 96 miles an hour for an incidence of —2° and a speed of about 47 miles an hour for 12°. The high speed is ample, but the low speed does not quite meet our specifications. However, it appears to be a better solution than any of the others. This is the price that must be paid for high speed in the air.
The above method of selecting a wing from its characteristic curves was suggested by M. Eiffel.9 In fact, the whole procedure of representing the properties of a wing in terms of its coefficients and incidence is due to this remarkable engineer. His method is applied here to a specific case.
6Data from Technical Report of Advisory Committee for Aeronautics, 1911-1912, London.
7Refer to Lanchester, "The Aeroplane from an Engineering Standpoint," Engineering, London, May 8, 1914.
8Data published in "Aerophile," Paris, May 15, 1914.
9“La Résistance de l'Air et Aviation," Paris, Dunot et Pinat, 1912.
Unfortunately, the intersection of the curves on Fig. 3 is very sharp and the results are at best approximate. The diagram is to be used only as a ready means of comparison to avoid the solution of the same problem six times to learn which of six wings is the best for a given problem. The performance of the aeroplane, climbing, etc., is not brought out for intermediate speeds on this plot. Later, a complete analysis of probable performance will be made.
Monoplane vs. Biplane.—The question of aspect ration, or ratio of span to depth of wing, has purposely not been introduced. The above characteristic curves are all drawn for an aspect ratio of six. It is found by experiment that with increase of aspect ratio Ky is increased for the same incidence and Kx very little altered. For a monoplane of 400 square feet area the span cannot be allowed to exceed 40 feet. This gives a depth of wing of 10 feet or an aspect ratio of four. A biplane of same span would have a depth of wing of five feet and an aspect ration of eight. The gain in lift in going from aspect ratio four to aspect ratio eight is in the neighborhood of 15 per cent in favor of the biplane. On the other hand, the ordinary biplane spacing of gap between wings equal to chord produces interference in the flow of air and a loss of lift of from 20 to 25 per cent under the equivalent monoplane of same aspect ratio and incidence. Recent experiments have shown however, that more than half of this loss may be made up by staggering the wings of the biplane and inclining the forward wing at a slightly greater angle than the other. The exact degree of staggering and difference in inclination must be worked out by experiments on models in an artificial current of air. It is assumed that such tests would be made before this design is completed. It is reasonable to conclude then that the values of the coefficients for an aspect ratio six, as used above, may safely be applied to our design if we use a biplane of aspect ratio eight, with a proper degree of stagger.10 We could even expect a slight margin of lift in our favor which later could be used to carry additional weight of gasoline.
10The Sopwith biplane which won the Monaco meet in May has wings staggered as above suggested.
Before leaving the question of wings, it may be of interest to remark that wing Eiffel No. 37 has a best ratio of lift to resistance of 20. This is then the most effective wing of which there are any published tests. The usual wing has a lift to resistance ratio of 15. Its curvature is shown on Fig. 4. On the same sheet are given the curves of Kx and Ky plotted on angle of chord to wind. These are for use in the subsequent calculation for resistance.
POWER AND RESISTANCE CURVES
Having determined the principal dimensions of our aeroplane and having some assurance that the specifications may be met, we wish now to analyze its probable performance and to design a propeller to satisfy the problem. The method of drawing resistance curves is that used at the Royal Air Craft Factory, England. We will compute for a series of speeds the resistance of the aeroplane, and hence the thrust the propeller must develop at each speed. Similarly, we will compute the effective power required at each speed which is equal to the propeller power that must be supplied.
We have computed above the value of Ky required at each speed to exert a lift of six pounds per square foot on the wings and hence to sustain the aeroplane. From the characteristic curves on Fig. 4 we can pick out the angle of incidence i corresponding to each of the values of Ky. We have then the inclination of the wing chord at each speed and hence have fixed the attitude of the machine. For each angle of incidence i there corresponds a resistance coefficient Kx which is picked off the Kx curve for the wing. The wing resistance is then computed for each speed by:
Rw=KxSV2.
The resistance of body, chassis, struts and wires does not in any way contribute to sustentation, but absorbs a great deal of power. Every effort will be made to reduce this to a minimum. From published tests on miscellaneous objects it is possible to compute the resistance of each part and then sum up to get the total passive or body resistance. The method is long and very approximate on account of rather arbitrary allowances that must be made for the sheltering of parts which are partly in the lee of other parts. The most trustworthy method is to make a model without wings at this stage in the design, and to test it in a wind tunnel. For want of a better guide, let us accept as true that we are unable to reduce the body resistances more than the Royal Air Craft Factory designers succeeded in doing with the BE-2. Tests on that aeroplane complete with fuselage, chassis, etc., but without wings, showed that the so-called body resistance is given by the equation:11
Rb=.01132V2.
We will then compute Rb for each speed and tabulate it below. The sum of wing resistance and body resistance is the total resistance to motion and must equal the propeller thrust:
T=Rw+Rb.
The product of thrust by velocity gives the effective horsepower
expended in driving the aeroplane:
P=TV/550.
11Since the above was written, the Technical Report of the Advisory Committee has appeared in which, on page 250, it is shown that the body resistance first estimated should be increased by 40 per cent to allow for the effect of the propeller race and to take account of more recent tests in the wind tunnel. The detailed estimate is given as follows and is reproduced as of particular interest to designers. The resistance of our aeroplane may then be increased 20 per cent and the maximum speed reduced to show 94 miles an hour. The minimum speed is unchanged.
BODY RESISTANCE OF BE-2 AEROPLANE AT 60 MILES AN HOUR
Struts: Pounds
8, 6' o" X 1¼" 4.2
4, 4' 0" X 1¼" 1.4
6, 3' o" x 1¼" 2.6
Wire:
220' cable 29.5
70' wire 5.6
50 turnbuckles 3.0
Rudder and elevator 2.0
Body with passenger and pilot 40.0
Axle 2.0
Main skids and axle mounting 2.0
Rear skid .5
Wheels 3.5
Appendages and miscellaneous 10.0
Parts exposed to slip stream of 25' per sec.: Body; 4, 4' 0" struts; 2/3' of 3’ 0" struts; 50’ cable; 50' wire; rudder and elevator rear skid, fittings, etc., calling for an augment of resistance of
35.7
Total 140.0
This quantity is tabulated for each speed in the following table:
The results of this calculation are shown graphically by a series of curves on Fig. 5, following the method of the Royal Air Craft Factory.
It appears from this plot that the body resistance which varies as the square of the speed plays an unimportant part at speeds below 55 miles an hour. In the first Wright aeroplanes of low speed, no effort was made to keep down body resistance, and the pilot and passenger were seated side by side on the lower wing entirely exposed to the wind. Above 75 miles an hour, the body resistance is greater than the wing resistance on account of the fact that the latter decreases very rapidly for the smaller incidence employed at high speed. The importance of minimizing body resistance for a high speed machine is obvious. In our design, motor, tanks, personnel and equipment will be completely sheltered inside a torpedo-shaped fuselage. Struts will be given a stream line section, wires will be replaced as far as possible by steel ribbons, and the cooling radiators carefully chosen and carefully located with a view to keeping head resistance low. In general, no appendages or projections will be allowed, all control wires being enclosed within the fuselage. The chassis will be made of steel tubing of egg-shaped section, having the blunt end forward. All struts and braces should have the rear edge fined by use of special sections or application of wooden "fairwaters." Wheels of the chassis have the spokes covered with fabric or sheet aluminum.
The total resistance curve shows a high maximum at low speed due to the great incidence of the wings required for sustentation. A high propeller thrust is therefore required.
The curve of effective horse-power is of greatest interest to the designer. This curve shows at every point the power required to drive the aeroplane. For flight to be possible, the propeller must be able to supply at least this power. From the relation,
Propeller power available = Thrust x Speed,
550
it follows that if the propeller delivers sufficient power it also delivers sufficient thrust to drive the aeroplane. The power required curve is then the only curve that must be considered.
In general, for any power there are two speeds at which the aeroplane can fly. For example, for 60 effective horse-power, the horizontal power line intersects the power required curve at two points corresponding to 46.5 and 80 miles an hour. The machine has the wing chord inclined 12° in the former case and —0.4° in the latter case. For any power between 41 and 60 there are two such flight speeds possible, corresponding to the two branches of the power required curve. French writers have indulged in considerable discussion with regard to the advantages of making use in practice of both branches of the curve and so obtaining a slow landing speed.12
12Alexandre Sée, "Les Lois Expérimentales de l'Aviation," Paris, Gauthier-Villars, 1912, page 200.
Capitaine du Gênie Duchene, "L'Aeroplane etudié et calculé par les Mathematiques Elementaires," Paris, Librairie Chapelot, 1913, page 116.
J. Bordeaux, "L'Aeroplane, Etude Raisonnée," Paris, Gauthier-Villars, 1912, page 223.
Suppose the aeroplane is flying at about 80 miles an hour with motor throttled down to give 60 propeller horse-power. The incidence of the wings is then —0.4°. If the pilot wishes to climb over a small hill, he will raise his elevator and throw the machine into a larger angle of incidence, for example, +1.2°. For this incidence, the machine will tend to slow down to 70 miles an hour, but at that speed only requires 47 horse-power. The propeller is still delivering approximately 60 horse-power or an excess of about 13 horse-power. This excess of power will be expended in lifting the machine as a whole, giving it a vertical component of velocity, calculated from
Excess power= Total weight x Climb per second
550
The machine thus takes an upward inclined- trajectory as desired.
Conversely, to descend, the pilot decreases his incidence to —1.45°, for example. At this incidence, a speed of 90 miles is necessary for sustentation and 80 horse-power. The propeller is only giving 60, and hence the necessary 20 horse-power must be supplied by the machines falling at a rate V feet per second given by:
20 H. P.= Total weight x V
550
The machine then speeds up and descends as desired.
In practice, the pilot need not know the change of incidence he produces. A careful man moves his elevator slowly until he has placed himself on the desired trajectory. Part of the art of aviation is to do this without exceeding safe limits, for obviously there is a limit to the rate of climb his motor can handle. If the machine is put on a climb too steep for the power of the machine, the speed is suddenly lost, the controls become ineffective and the machine has "stalled." It may side slip, settle back on its tail and in response to the pilot's efforts to level up may take a steep dive. This all is dangerous, especially if the ground is too near.
To consider the left branch of the power curve, called the “régime lent," as distinguished from the "régime rapide," suppose the pilot by a suitable manipulation of controls has put his machine at an angle 8.15° and is flying at 50 miles with 49 horse-power. If he wishes to rise, he naturally puts the machine as before at a greater angle, say 12°. The machine slows down to 46.5 miles. At that speed and angle 60 horse-power is required. His propeller is only giving him 49 horse-power and the aeroplane will descend at a sufficient rate to make up the difference. On the other hand, if he wishes to descend and so reduces his incidence to 3.65°, the aeroplane tends to speed up to 60 miles. For this speed only 41 horse-power is required and the machine will climb at a rate to about 8 horse-power, excess power given out by the propeller. Consequently, his controls are reversed, when he heads down he rises, and when he heads up he descends, provided he does not manipulate his throttle. For a pilot who is n6t familiar with this abnormal behavior of his machine, the "régime lent" is unsafe and he should not attempt to fly at speeds lower than 60 miles an hour. To avoid danger, an air speed indicator should be before his eyes so that he may be warned before losing too much speed.
It has been recently demonstrated that an expert flyer can operate an aeroplane in the slow speed region by manipulation of the throttle alone. The recent speed variation contests abroad have placed a premium on low speed and formed the inducement to pilots to attempt the region of reversed controls. The usual method is to observe by the speed indicator that the aeroplane has passed below its critical speed and then to control elevation by use of the throttle alone. Thus to descend, power is shut off, and to rise more gas is turned on. A flexible motor is, of course, very necessary. With a large reserve of power, an aeroplane can be made to climb out of any difficulty. The matter of low speed is a question of skill of the pilot, and is dangerous when not understood.
Limits of speed and climb, being primarily based on power available, it remains now to consider the calculation of this quantity at all speeds. At the present time, no satisfactory method has been developed to determine in advance the propeller performance at all speeds. In fact, the whole method of propeller design is largely a matter of trial and error in an effort to hit on a propeller suitable to the motor and to the aeroplane. It is even urged that the best method is to try several propellers and to adopt that one which gives the greatest pull when the aeroplane is held to a stake by a spring scale.
The problem differs in no way from the propulsion of a ship except that there is less information available derived from experience.
A gasoline motor develops its full power at one speed of revolution and at no other. The power falls off directly as the revolutions. The propeller must then turn at a fixed rate in order not to handicap the motor. On the other hand, at any speed the aeroplane has a definite resistance which must be overcome by the propeller thrust. The slip of the propeller, if run at constant revolutions, increases with slower aeroplane speeds. The efficiency of the propeller is a maximum at one slip and less for values of slip above and below this figure. It is a question of judgment to decide for what speed of the aeroplane the propeller efficiency shall be a maximum.
From a consideration of the effective power required, as given by Fig. 5, it appears that at high speeds we need all the power we can get. At low speeds, an inefficient propeller will probably give power enough.
Let us then design our propeller to give its maximum efficiency at 90 miles an hour. The Salmson motor develops 125 horsepower at 1300 revolutions per minute, which must be the propeller speed. To develop maximum power, we will run the propeller at 1300 revolutions per minute for all speeds of the aeroplane.
The problem then is to design a propeller to absorb 125 horse-power at 1300 revolutions per minute and to develop a thrust of 380 pounds at 90 miles an hour. The resistance of the aeroplane is only 340 pounds at 90 miles, so we are allowing a margin for safety of 40 pounds thrust.
The rational method of propeller design devised by M. Stefan Drzwiecki can be applied now, and a propeller designed with the assurance that it will come within 5 or 10 per cent of the calculated values. We have a tolerance above of 10 per cent. It is of interest to note that the Drzwiecki theory is more often found to underestimate than to overestimate. The detailed method is described by M. Drzwiecki13 and further amplified by the graphical method used by the Royal Air Craft Factory.14
The method is long and involves the graphical integration of the pressures over the propeller blade. The detailed design of a propeller cannot well be attempted here.
An application of the theory of similitude is developed in Appendix I, with a view to making use of M. Drzwiecki's propellers tested full size on the electric railway of the University of Paris.
Three propellers are found to give an efficiency of 80 per cent—a remarkable performance. Not to burden the body of this paper, we will extract the data obtained by the process explained in the Appendix.
13Des Hélices Aériennes," Theorie Générale des Propulseurs heli-coidaux, S. Drzwiecki, Paris, F. Louis Vivian, 1909.
14Technical Report of the Advisory Committee for Aeronautics for the year 1911-1912, London, His Majesty's Stationery Office, Report No. 65 by Mervyn O'Gorman, Superintendent, Royal Air Craft Factory.
Propeller prototype, St. Cyr No. 8.15 Revolutions per minute, 1300. Diameter, 7.55 feet. Two blades. Material, walnut. Power absorbed from motor, 125 horse-power.
Miles per Hour Thrust Horse-Power
45 71.2
50 78.0
60 86.8
70 92.5
80 96.8
90 100.0
100 97.5
The thrust or propeller horse-power as given above is plotted as a curve of power available on Fig. 5. Flight is possible so long as the power required curve does not exceed the power available. It is seen that the maximum horizontal speed is about 98 miles an hour. The slowest speed is given by the speed at which the planes lose their lift, 46.5 miles an hour, and is not limited by the propeller power. At 46.5 miles an hour there is 13 horse-power excess available for climbing. It is, therefore, practicable to fly the machine at 46.5 miles an hour. This may be taken as the minimum landing speed. The aeroplane also could leave the ground at this speed if the pilot were careful to reduce his inclination as soon as he was clear of the ground. This would increase his speed to 60 miles, and give an excess power of 45 horse-power with full throttle opening tending to make the machine climb rapidly to clear nearby fences and trees.
In flight, speeds between 60 and 98 miles an hour are available by throttling down. At the maximum speed, the radius of action of six hours gives a flight of 588 miles. This is based on the weight of the machine with full tanks. As time goes on, the machine is lightened by consumption of gasoline and flies at a smaller incidence with less power. The radius of action is therefore greater than 588 miles. Account of this will be taken later.
The greatest margin of power is 47 horse-power at 65 miles an hour. This is the difference between ordinates of the two power curves. Consequently, 65 miles an hour is the best horizontal speed for climbing. A rate of climb with full tanks is then:
V=47 X 33,00/2387=650 feet per minute.
This is well over our specification.
15Bulletin du Laboratoire Aérotechnique de St. Cyr, University of Paris, 1913.
The minimum power to fly is 41 horse-power at 60 miles an hour. With a motor missing fire badly, the pilot should regulate his speed to 60 miles if possible until a landing can be made.
It is shown in the Appendix how to estimate the variation in propeller power with speed for full throttle opening, as well as for reduced power. The variation of propeller efficiency and the consequent propeller power are calculated for 1300, 1000, 900, 800, 700 and 600 revolutions.
On Fig. 6, the effective power required curve is again laid down and across it are plotted curves of propeller horse-power for various throttle openings. It is seen that the speed of the aeroplane for maximum propeller efficiency at reduced power falls off so rapidly that, in general, the propeller is working at excessive slip and consequent reduced efficiency. Part of the saving of fuel by running at reduced power is therefore lost in the propeller.
The most economical speed is that speed which will give the greatest radius of action. The fuel tank capacity is constant so that the duration of the flight in hours is proportional to:
Tank Capacity,
B.H.P.
assuming the same economy of the motor at full and reduced power. The motor is likely to be more economical at reduced power. The distance flown is proportional to velocity times time, and since tankage is constant, varies as………….
Velocity.
B.H.P.
We must then find a curve of brake horse-power referred to V as abscissæ.
The radius of action is naturally dependent on weight, so allowance must first be made for the lightening of the aeroplane as fuel is consumed.
Take first .the condition at the end of the flight with tanks practically empty. The machine is lightened by 483 pounds of fuel and oil, giving a total weight of 1914 pounds. The wing loading is now 4.82 pounds per square foot of wing area. The power and resistance curves can now be constructed exactly as in the preceding case for full tanks.
The work is tabulated below. For sake of convenience, only the curve of effective horse-power required for empty tanks is plotted on Fig. 6.
A comparison of the effective horse-power curves for full and empty tanks shows a marked lowering of power required at low speeds and a lowering of the minimum speed at which flight is possible from 46.5 to 41.6 miles an hour. This is very satisfactory, in view of the specified minimum of 45 miles.
The reduction of effective horse-power as the fuel tanks become lighter gives little saving in brake horse-power required from the motor on account of propeller losses. The curves of brake horse-power for various openings of the throttle fall off very rapidly from a maximum. This is a result of our having elected to design our propeller for maximum efficiency at 90 miles an hour. We might have done better to design for 100 miles an hour and always work at a little below the best efficiency. In that case, the radius of action at reduced power could be augmented.
Let us now pick off the speed corresponding to each brake horse-power motor curve on Fig. 6 where it intersects the effective horse-power curve. We then know the brake horse-power required for each speed of the aeroplane. The mean of values taken for tanks full and empty is plotted as a curve of brake horse-power on velocity. (See Fig. 7.)
We have shown above that the most economical speed gives a maximum for the ratio V/B.H.P. As is in the case of ships, we can find the maximum of this fraction, by drawing a tangent from the origin to the brake horse-power curve. The dotted line on Fig. 7 is tangent to the curve at about 70 miles an hour, at which speed a brake horse-power of 73 horse-power must be developed by the motor to propel the aeroplane. The effective horse-power required for this speed is about 47 horse-power so the propeller is working at only 64 per cent efficiency. The duration of flight at 70 miles an hour is:
6 X 125/73 = 10.25 hours,
and the total distance:
10.25 X 70 = 717 miles.
This compares with a radius of action at full power of six hours at 98 miles an hour, or 588 miles.
It is worth while again to point out the necessity of an air speed indicator. If the longest flight can be made at 70 miles, it is obvious that the pilot must adjust his speed to 70 miles. If his motor is in good condition, he knows that if it turns over 800 times per minute his air speed is 70 miles an hour. This indication is only a rough approximation, however.
In flight since the lift, head resistance, thrust and weight form a closed polygon of forces, the cutting off of power leaves the machine under the influence of weight, lift and resistance. These forces form a triangle of forces and draw the aeroplane along an inclined trajectory. The pitch or angle of glide is such that the projection of the weight on the line of the trajectory is equal to the resistance to motion, and the projection, normal to the trajectory, is equal to the lift of the planes in the same line. The tangent of the angle of glide is therefore the ratio between resistance and lift. The best gliding angle can be estimated by reference to Fig. 5. The minimum resistance with full tanks is 250 pounds at 65 miles an hour. The lift is equal to the weight. Hence, the pitch of glide is about 1 in 9.6. This is better than the average of modern aeroplanes where 1 in 8 is considered sufficient. With tanks empty, the minimum resistance is 205 pounds at 60 miles, and the lift or weight 1914. The glide is then about 1 in 9.35.
Experiments with a Bleriot monoplane at St. Cyr indicate that with power cut off and propeller turning idle during a glide, the propeller acts as a brake and in some cases may increase the total resistance as much as 20 per cent. This would give us a glide with full tanks of about 1 in 7.7. The gliding angle is not difficult to determine in the completed machine.
The aeroplane as above studied promises to meet our very rigid specifications and is worth proceeding with the detailed design. As the structural elements are designed the weight is determined carefully, and the speed and power curves corrected. When plans are complete, model tests of body and chassis with radiator and appendages will be made to verify the estimate of body resistance. Further correction of the characteristic curves can then be made, and if the results still promise success, the construction of a full scale aeroplane is undertaken. After extensive trials, the characteristic curves as calculated may be verified at a few points. For this, it is necessary to measure propeller thrust in flight, together with speed and incidence.
The above aeroplane design then should have the following approximate characteristics:
Total weight, bare, 1598 lbs.
Oil and gasoline, 482 lbs.
Two men, 320 lbs.
Total weight loaded, 2400 lbs.
Area of wings, 400 sq. ft.
Wing loading in pounds per square foot, 6 lbs.
Type of wings, Eiffel No. 37.
Arrangement, staggered biplane.
Span, 40 ft. Depth, 5 ft. Aspect ratio, 8.
Gap between wings, 5 ft.
Length of body, 20 ft.
Motor: 125 horse-power Salmson (Canton Unné) (Type M-9, 9 cylinders, bore 120 mm., stroke 130 mm.).
Propeller: two-bladed, diameter 7.54 ft.
Type of propeller: St. Cyr No. 8.
Type of chassis: 4 wheels; steel tube.
Maximum speed, 98 miles an hour.
Landing speed: full tanks, 46.5 miles an hour; empty tanks, 41.5 hour.
Radius at full power: 6 hours, 588 miles.
R. P. M. at full power, 1300.
Economical speed, 70 miles an hour.
Radius at economical speed: 10.25 hours, 717 miles.
B. H. P. at 70 miles: 73 horse-power, 800 R. P. M.
Theoretical best gliding angle, 1/9.6.
Speed at best gliding angle, 65 miles an hour.
Initial rate of climb: full tanks, 650 ft. per min.; empty tanks, 928 ft. per min.
Horizontal speed of climb, 65 miles an hour.
Propeller efficiency at 90 miles, 80%.
Propeller efficiency at 70 miles, 64%.