§ 6. It will not do to tip up the elevator to a large angle to make it rise quickly, because when once off the ground the angle of the elevator is wrong for actual flight and the model will probably turn a somersault and land on its back. I have often seen this happen. If the elevator be set at an increased angle to get it to rise quickly, then what is required is a little mechanical device which sets the elevator at its proper flight angle when it leaves the ground. Such a device does not present any great mechanical difficulties; and I leave it to the mechanical ingenuity of my readers to devise a simple little device which shall maintain the elevator at a comparatively large angle while the model is on the ground, but allowing of this angle being reduced when free flight is commenced.

§ 7. The propeller most suitable to "get the machine off the ground" is one giving considerable statical thrust. A small propeller of fine pitch quickly starts a machine, but is not, of course, so efficient when the model is in actual flight. A rubber motor is not at all well adapted for the purpose just discussed.

§ 8. Professor Kress uses a polished plank (down which the models slip on cane skids) to launch his models.

§ 9. When launching a twin-screw model the model should be held by each propeller, or to speak more correctly, the two brackets holding the bearings in which the propeller shafts run should be held one in each hand in such a way, of course, as to prevent the propellers from revolving. Hold the machine vertically downwards, or, if too large for this, allow the nose to rest slightly on the ground; raise (or swing) the machine up into the air until a little more than horizontal position is attained, and boldly push the machine into the air (moving forward if necessary) and release both brackets and screws simultaneously.[46]

§ 10. In launching a model some prefer to allow the propellers to revolve for a few moments (a second, say) before actually launching, contending that this gives a steadier initial flight. This is undoubtedly the case, see note on page 111.

§ 11. In any case, unless trying for a height prize, do not point the nose of the machine right up into the air with the idea that you will thereby obtain a better flight.

Launch it horizontally, or at a very small angle of inclination. When requiring a model to run along a field or a lawn and rise therefrom this is much facilitated by using a little strip of smooth oilcloth on which it can run. Remember that swift flying wooden and metal models require a high initial velocity, particularly if of large size and weight. If thrown steadily and at the proper angle they can scarcely be overthrown.

CHAPTER XI.

§ 3. In 1842 a Mr. Phillips constructed a successful power-driven model helicopter. The model was made entirely of metal, and when complete and charged weighed 2 lb. It consisted of a boiler or steam generator and four fans supported between eight arms. The fans had an inclination to the horizon of 20°, and through the arms the steam rushed on the principle of Hero's engines (Barker's Mill Principle probably). By the escape of steam from the arms the fans were caused to revolve with immense energy, so much so that the model rose to an immense altitude and flew across two fields before it alighted. The motive power employed was obtained from the combustion of charcoal, nitre and gypsum, as used in the original fire annihilator; the products of combustion mixing with water in the boiler and forming gas-charged steam, which was delivered at high pressure from the extremities of the eight arms.[47] This model and its flight (fully authenticated) is full of interest and should not be lost sight of, as in all probability being the first model actuated by steam which actually flew.

The helicopter is but a particular phase of the aeroplane.

§ 4. The simplest form of helicopter is that in which the torque of the propeller is resisted by a vertical loose fabric plane, so designed as itself to form a propeller, rotating in the opposite direction. These little toys can be bought at any good toy shop from about 6d. to 1s. Supposing we desire to construct a helicopter of a more ambitious and scientific character, possessing a vertically rotating propeller or propellers for horizontal propulsion, as well as horizontally rotating propellers for lifting purposes.

§ 5. There is one essential point that must be carefully attended to, and that is, that the horizontal propulsive thrust must be in the same plane as the vertical lift, or the only effect will be to cause our model to turn somersaults. I speak from experience.

When the horizontally revolving propellers are driven in a horizontal direction their "lifting" powers will be materially increased, as they will (like an ordinary aeroplane) be advancing on to fresh undisturbed air.

§ 6. I have not for ordinary purposes advocated very light weight wire framework fabric-covered screws, but in a case like this where the thrust from the propeller has to be more than the total weight of the machine, these might possibly be used with advantage.

§ 7. Instead of using two long vertical rods as well as one long horizontal one for the rubber strands, we might dispense with the two vertical ones altogether and use light gearing to turn the torque action through a right angle for the lifting screws, and use three separate horizontal rubber strands for the three propellers on a suitable light horizontal framework. Such should result in a considerable saving of weight.

§ 8. The model would require something in the nature of a vertical fin or keel to give the sense of direction. Four propellers, two for "lift" and two for "drift," would undoubtedly be a better arrangement.

CHAPTER XII.

A model flying machine being a scientific invention and not a toy, every devotee to the science should make it his or her business to keep, as far as they are able, accurate and scientific records. For by such means as this, and the making known of the same, can a science of model aeroplaning be finally evolved. The following experimental entry forms, left purposely blank to be filled in by the reader, are intended as suggestions only, and can, of course, be varied at the reader's discretion. When you have obtained carefully established data, do not keep them to yourself, send them along to one of the aeronautical journals. Do not think them valueless; if carefully arranged they cannot be that, and may be very valuable.

Experimental Data.
Form I.

Form I.continued.

CHAPTER XIII.

1. Longest flight measured in a straight line (sometimes both with and against the wind).[48]

2. Stability (both longitudinal and transverse).

3. Longest glide when launched from a given height without power, but with motor and propeller attached.

4. Steering.

5. Greatest height.

6. The best all-round model, including, in addition to the above, excellence in building.

Generally so many "points" or marks are given for each test, and the model whose aggregate of points makes the largest total wins the prize; or more than one prize may be offered—

One for the longest flight.

One for the swiftest flight over a measured distance.

One for the greatest height.

One for stability and steering.

And one for the best all-round model.

The models are divided into classes:—

§ 2. Aero Models Association's Classification, etc.

All surfaces, whether vertical, horizontal, or otherwise, to be calculated together for the above classification.

All round efficiency—marks or points as percentages:—

Two prizes:—

One for length of flight.

One for all-round efficiency (marked as above).

Every competitor to be allowed three trials in each competition, the best only to count.

All flights to be measured in a straight line from the starting to the landing point.

Repairs may be made during the competition at the direction of the judges.[50]

There are one or two other points where flights are not made with and against the wind. The competitors are usually requested to start their models from within a given circle of (say) six feet diameter, and fly them in any direction they please.

"Gliding angle" means that the model is allowed to fall from a height (say) of 20 ft.

"Directional control," that the model is launched in some specified direction, and must pass as near as possible over some indicated point.

The models are practically always launched by hand.

§ 3. Those who desire to win prizes at such competitions would do well to keep the following points well in mind.

1. The distance is always measured in a straight line. It is absolutely essential that your model should be capable of flying (approximately) straight. To see, as I have done, model after model fly quite 150 to 200 yards and finish within 50 yards of the starting-point (credited flight 50 yards) is useless, and a severe strain on one's temper and patience.

2. Always enter more than one model, there nearly always is an entrance fee; never mind the extra shilling or so. Go in to win.

3. It is not necessary that these models should be replicas of one another. On some days a light fabric-covered model might stand the best chance; on another day, a swift flying wooden or metal aerofoil.

Against the wind the latter have an immense advantage; also if the day be a "gusty" one.[51]

4. Always make it a point of arriving early on the ground, so that you can make some trial flights beforehand. Every ground has its local peculiarities of air currents, etc.

5. Always be ready in time, or you may be disqualified. If you are flying a twin-screw model use a special winder, so that both propellers are wound up at the same time, and take a competent friend with you as assistant.

6. For all-round efficiency nothing but a good all-round model, which can be absolutely relied on to make a dozen (approximately) equivalent flights, is any good.

7. In an open distance competition, unless you have a model which you can rely on to make a minimum flight of 200 yards, do not enter unless you know for certain that none of the "crack" flyers will be present.

8. Do not neglect the smallest detail likely to lead to success; be prepared with spare parts, extra rubber, one or two handy tools, wire, thread, etc. Before a lecture, that prince of experimentalists, Faraday, was always careful to see that the stoppers of all the bottles were loose, so that there should be no delay or mishap.

9. If the rating of the model be by "weight" (1 oz., 2 oz., 4 oz., etc.) and not area, use a model weighing from 10 oz. to a pound.

10. If there is a greatest height prize, a helicopter model should win it.[52] (The writer has attained an altitude of between three and four hundred feet with such.) The altitude was arrived at by observation, not guesswork.

11. It is most important that your model should be able to "land" without damage, and, as far as possible, on an even keel; do not omit some form of "skid" or "shock-absorber" with the idea of saving weight, more especially if your model be a biplane, or the number of flights may be restricted to the number "one."

12. Since the best "gliding" angle and "flying" angle are not the same, being, say, 7° in the former case and 1°-3°, say, in the latter, an adjustable angle might in some cases be advantageous.

13. Never turn up at a competition with a model only just finished and practically untested which you have flown only on the morning of the competition, using old rubber and winding to 500 turns; result, a flight of 250 yards, say. Arrived on the competition ground you put on new rubber and wind to 750 turns, and expect a flight of a quarter of a mile at least; result 70 yards, measured in a straight line from the starting-point.

14. Directional control is the most difficult problem to overcome with any degree of success under all adverse conditions, and 15 per cent., in the writer's opinion, is far too low a percentage; by directional I include flying in a straight line; personally I would mark for all-round efficiency: (A) distance and stability, 50 per cent.; (B) directional control, 30 per cent.; (C) duration of flight, 20 per cent. In A the competitor would launch his model in any direction; in B as directed by the judges. No separate flights required for C.

CHAPTER XIV.

§ 3. Total surface of a cylinder = circumference of base × height + 2 area of base.

Area of a circle = square of diameter × 0·7854.

Area of a circle = square of rad. × 3·14159.

Area of an ellipse = product of axes × 0·7854.

Circumference of a circle = diameter × 3·14159.

Solidity of a cylinder = height × area of base.

Area of a circular ring = sum of diameters × difference of diameters × 0·7854.

For the area of a sector of a circle the rule is:—As 360 : number of degrees in the angle of the sector :: area of the sector : area of circle.

To find the area of a segment less than a semicircle:—Find the area of the sector which has the same arc, and subtract the area of the triangle formed by the radii and the chord.

The areas of corresponding figures are as the squares of corresponding lengths.

§ 4.

§ 5. One atmosphere = 14·7 lb. per sq. in. = 2116 lb. per sq. ft. = 760 millimetres of mercury.

A column of water 2·3 ft. high corresponds to a pressure of 1 lb. per sq. in.

1 H.P. = 33,000 ft.-lb. per min. = 746 watts.

Volts × amperes = watts.

π = 3·1416.    g = 32·182 ft. per sec. at London.

§ 6. Table of Equivalent Inclinations.

§ 7. Table of Skin Friction.
Per sq. ft. for various speeds and surface lengths.

This table is based on Dr. Zahm's experiments and the equation

f = 0·00000778l ^-0·07v^1·85

Where f = skin friction per sq. ft.; l = length of surface; v = velocity in feet per second.

In a biplane model the head resistance is probably from twelve to fourteen times the skin friction; in a racing monoplane from six to eight times.

§ 8. Table I.—(Metals).

[A] E in millions of lb. per sq.in.
[B] Magnalium is an alloy of magnesium and aluminium.
[C] Aluminium 94 per cent., copper 6 per cent. (the best percentage), a 6 per cent. alloy thereby doubles the tenacity of pure aluminium with but 5 per cent. increase of density.

§ 9. Table II.—Wind Pressures.

p = kv².

k coefficient (mean value taken) ·003 (miles per hour) = 0·0016 ft. per second. p = pressure in lb. per sq. ft. v = velocity of wind.

§ 10. Representing normal pressure on a plane surface by 1; pressure on a rod (round section) is 0·6; on a symmetrical elliptic cross section (axes 2:1) is 0·2 (approx.). Similar shape, but axes 6:1, and edges sharpened (see ch. ii., § 5), is only 0·05, or 1/20, and for the body of minimum resistance (see ch. ii., § 4) about 1/24.

§ 11. Table III.—Lift and Drift.

On a well shaped aerocurve or correctly designed cambered surface. Aspect ratio 4·5.

Wind velocity 40 miles per hour. (The above deduced from some experiments of Sir Hiram Maxim.)

At a velocity of 30 miles an hour a good aerocurve should lift 21 oz. to 24 oz. per sq. ft.

§ 12. Table IV.—Lift and Drift.

On a plane aerofoil.

N = P(^{2 sin α}/_{1 + sin² α})

P = 2kd AV² sin α.

A useful formula for a single plane surface. P = pressure supporting the plane in pounds per square foot, k a constant = 0·003 in miles per hour, d = the density of the air.

A = the area of the plane, V relative velocity of translation through the air, and α the angle of flight.

Transposing we have

AV² = P/(2kd sin α)

If P and α are constants; then AV² = a constant or area is inversely as velocity squared. Increase of velocity meaning diminished supporting surface (and so far as supporting surface goes), diminished resistance and skin friction. It must be remembered, however, that while the work of sustentation diminishes with the speed, the work of penetration varies as the cube of the speed.

§ 13. Table V.—Timber.

t = thickness: d = diameter.
[A]Given elsewhere as 55 and 22,500 (t = 1/3 d), evidently regarded as solid.

§ 14.—Formula connecting the Weight Lifted in Pounds per Square Foot and the Velocity.—The empirical formula

W = (V²C)/g

Where W = weight lifted in lb. per sq. ft.
V  = velocity in ft. per sec.
C  = a constant = 0·025.
g = 32·2, or 32 approx.

may be used for a thoroughly efficient model. This gives (approximately)

Remember the results work out in feet per second. To convert (approximately) into miles per hour multiply by 2/3.

§ 15. Formula connecting Models of Similar Design, but Different Weights.

D ∝√W.

or in models of similar design the distances flown are proportional to the square roots of the weights. (Derived from data obtained from Clarke's flyers.)

For models from 1 oz. to 24-30 oz. the formula appears to hold very well. For heavier models it appears to give the heavier model rather too great a distance.

Since this was deduced a 1 oz. Clarke model of somewhat similar design but longer rubber motor has flown 750 ft. at least; it is true the design is not, strictly speaking, similar, but not too much reliance must be placed on the above. The record for a 1 oz. model to date is over 300 yards (with the wind, of course), say 750 ft. in calm air.

§ 16. Power and Speed.—The following formula, given by Mr. L. Blin Desbleds, between these is—

^W/_W0 = ^3v₀/_4v + ¼(^v/_v0)³.

Where v₀ = speed of minimum power
W₀ = work done at speed v₀.
W = work done at speed v.

Making v = 2v₀, i.e. doubling the speed of minimum power, and substituting, we have finally

W = (2³/₈)W₀

i.e. the speed of an aeroplane can be doubled by using a power 2³/₈ times as great as the original one. The "speed of minimum power" being the speed at which the aeroplane must travel for the minimum expenditure of power.

§ 17. The thrust of the propeller has evidently to balance the

Aerodynamic resistance = R
The head resistance (including skin friction) = S

Now according to Renard's theorem, the power absorbed by R + S is a minimum when

S = ^R/₃.

Having built a model, then, in which the total resistance

= ⁴/₃R.

This is the thrust which the propeller should be designed to give. Now supposing the propeller's efficiency to be 80 per cent., then P—the minimum propulsion power

= ⁴/₃R × ¹⁰⁰/₈₀ × ¹⁰⁰/₇₅ × v.

Where 25 per cent. is the slip of the screw, v the velocity of the aeroplane.

§ 18. To determine experimentally the Static Thrust of a Propeller.—Useful for models intended to raise themselves from the ground under their own power, and for helicopters.

The easiest way to do this is as follows: Mount the propeller on the shaft of an electric motor, of sufficient power to give the propeller 1000 to 1500 revolutions per minute; a suitable accumulator or other source of electric energy will be required, a speedometer or speed counter, also a voltmeter and ammeter.

Place the motor in a pair of scales or on a suitable spring balance (the former is preferable), the axis of the motor vertical, with the propeller attached. Rotate the propeller so that the air current is driven upwards. When the correct speed (as indicated by the speed counter) has been attained, notice the difference in the readings if a spring balance be used, or, if a pair of scales, place weights in the scale pan until the downward thrust of the propeller is exactly balanced. This gives you the thrust in ounces or pounds.

Note carefully the voltage and amperage, supposing it is 8 volts and 10 amperes = 80 watts.

Remove the propeller and note the volts and amperes consumed to run the motor alone, i.e. to excite itself, and overcome friction and air resistance; suppose this to be 8 volts and 2 amperes = 16; the increased load when the propeller is on is therefore

80 - 16 = 64 watts.

All this increased power is not, however, expended on the propeller.

The lost power in the motor increases as C²R.

R = resistance of armature and C = current. If we deduct 10 per cent. for this then the propeller is actually driven by 56 watts.

Now 746 watts = 1 h.p.

∴ ⁵⁶/₇₄₆ = ¹/₁₃ h.p. approx.

at the observed number of revolutions per minute.

§ 19. N.B.—The h.p. required to drive a propeller varies as the cube of the revolutions.

Proof.—Double the speed of the screw, then it strikes the air twice as hard; it also strikes twice as much air, and the motor has to go twice as fast to do it.

§ 20. To compare one model with another the formula

^{Weight × velocity (in ft. per sec.)}/_horse-power

is sometimes useful.

§ 21. A Horse-power is 33,000 lb. raised one foot in one minute, or 550 lb. one foot in one second.

A clockwork spring raised 1 lb. through 4½ ft. in 3 seconds. What is its h.p.?

1 lb. through 4½ ft. in 3 seconds
is 1 lb. " 90 ft. " 1 minute.

∴ Work done is 90 ft.-lb.
= ⁹⁰/₃₃₀₀₀ = 0·002727 h.p.

The weight of the spring was 6¾ oz. (this is taken from an actual experiment), i.e. this motor develops power at the rate of 0·002727 h.p. for 3½ seconds only.

§ 22. To Ascertain the H.P. of a Rubber Motor. Supposing a propeller wound up to 250 turns to run down in 15 seconds, i.e. at a mean speed of 1200 revolutions per minute or 20 per second. Suppose the mean thrust to be 2 oz., and let the pitch of the propeller be 1 foot. Then the number of foot-pounds of energy developed

= ^{2 oz. × 1200 revols. × 1 ft. (pitch)} / _{16 oz.}

= 150 ft.-lb. per minute.

But the rubber motor runs down in 15 seconds.
∴ Energy really developed is

= ^{150 × 15} / ₆₀ = 37·5 ft.-lb.

The motor develops power at rate of ¹⁵⁰/₃₃₀₀₀ = 0·004545 h.p., but for 15 seconds only.

§ 23. Foot-pounds of Energy in a Given Weight of Rubber (experimental determination of).

12 oz. were raised 19 ft. in 5 seconds.
i.e. ¾ lb. was raised 19 × 12 ft. in 1 minute.
i.e. 1 lb. was raised 19 × 3 × 3 ft. in 1 minute.
= 171 ft. in 1 minute.

i.e. 171 ft.-lb. of energy per minute. But actual time was 5 seconds.

∴ Actual energy developed by 2-7/16 oz. of rubber of 36 yards, i.e. 36 strands 1 yard each at 200 turns is

= ¹⁷¹/₁₂ ft.-lb.
= 14¼ ft.-lb.

This allows nothing for friction or turning the axle on which the cord was wound. Ball bearings were used; but the rubber was not new and twenty turns were still unwound at the end of the experiment. Now allowing for friction, etc. being the same as on an actual model, we can take ¾ of a ft.-lb. for the unwound amount and estimate the total energy as 15 ft.-lb. as a minimum. The energy actually developed being at the rate of 0·0055 h.p., or ¹/₂₀₀ of a h.p. if supposed uniform.

§ 24. The actual energy derivable from 1 lb. weight of rubber is stated to be 300 ft.-lb. On this basis 2-⁷/₁₆ oz. should be capable of giving 45·7 ft.-lb. of energy, i.e. three times the amount given above. Now the motor-rubber not lubricated was only given 200 turns—lubricated 400 could have been given it, 600 probably before rupture—and the energy then derivable would certainly have been approximating to 45 ft.-lb., i.e. 36·25. Now on the basis of 300 ft.-lb. per lb. a weight of ½ oz. (the amount of rubber carried in "one-ouncers") gives 9 ft.-lb. of energy. Now assuming the gliding angle (including weight of propellers) to be 1 in 8; a perfectly efficient model should be capable of flying eight times as great a distance in a horizontal direction as the energy in the rubber motor would lift it vertically. Now 9 ft.-lb. of energy will lift 1 oz. 154 ft. Therefore theoretically it will drive it a distance (in yards) of

^{8 × 154}/₃ = 410·6 yards.

Now the greatest distance that a 1 oz. model has flown in perfectly calm air (which never exists) is not known. Flying with the wind 500 yards is claimed. Admitting this what allowance shall we make for the wind; supposing we deduct half this, viz. 250 yards. Then, on this assumption, the efficiency of this "one ouncer" works out (in perfectly still air) at 61 per cent.

The gliding angle assumption of 1 in 8 is rather a high one, possibly too high; all the writer desires to show is the method of working out.

Mr. T.W.K. Clarke informs me that in his one-ouncers the gliding angle is about 1 in 5.

§ 25. To Test Different Motors or Different Powers of the Same Kind of Motor.—Test them on the same machine, and do not use different motors or different powers on different machines.

§ 26. Efficiency of a Model.—The efficiency of a model depends on the weight carried per h.p.

§ 27. Efficiency of Design.—The efficiency of some particular design depends on the amount of supporting surface necessary at a given speed.

§ 28. Naphtha Engines, that is, engines made on the principle of the steam engine, but which use a light spirit of petrol or similar agent in their generator instead of water with the same amount of heat, will develop twice as much energy as in the case of the ordinary steam engine.

§ 29.Petrol Motors.

§ 30. The Horse-power of Model Petrol Motors.—Formula for rating of the above.

(R.P.M. = revolutions per minute.)
H.P. = ^{(Bore)2 × stroke × no. of cylinders × R.P.M.}/_12,000

If the right-hand side of the equation gives a less h.p. than that stated for some particular motor, then it follows that the h.p. of the motor has been over-estimated.