Page:EB1911 - Volume 01.djvu/296

by Joessel’s formula, C+(0·2+0·3 sin α)L, in which C is the distance from the front edge, L the length fore and aft, and α the angle of incidence. The movement is different on concave surfaces. The term aeroplane is understood to apply to flat sustaining surfaces, but experiment indicates that arched surfaces are more efficient. S. P. Langley proposed the word aerodrome, which seems the preferable term for apparatus with wing-line surfaces. This is the type to which results point as the proper one for further experiments. With this it seems probable that, with well-designed apparatus, 40 to 50 ℔ can be sustained per indicated h.p., or about twice that quantity per resistance or “thrust” h.p., and that some 30 or 40% of the weight can be devoted to the machinery, thus requiring motors, with their propellers, shafting, supplies, &c., weighing less than 20 ℔ per h.p. It is evident that the apparatus must be designed to be as light as possible, and also to reduce to a minimum all resistances to propulsion. This being kept in view, the strength and consequent section required for each member may be calculated by the methods employed in proportioning bridges, with the difference that the support (from air pressure) will be considered as uniformly distributed, and the load as concentrated at one or more points. Smaller factors of safety may also have to be used. Knowing the sections required and unit weights of the materials to be employed, the weight of each part can be computed. If a model has been made to absolutely exact scale, the weight of the full-sized apparatus may approximately be ascertained by the formula

${\mbox{W}}'={\mbox{W}}{\sqrt {\left({\frac {{\mbox{S}}'}{\mbox{S}}}\right)^{3}}},$

in which W is the weight of the model, S its surface, and W′ and S′ the weight and surface of the intended apparatus. Thus if the model has been made one-quarter size in its homologous dimensions, the supporting surfaces will be sixteen times, and the total weight sixty-four times those of the model. The weight and the surface being determined, the three most important things to know are the angle of incidence, the “lift,” and the required speed. The fundamental formula for rectangular air pressure is well known: P＝KV²S, in which P is the rectangular normal pressure, in pounds or kilograms, K a coefficient (0·0049 for British, and 0·11 for metric measures), V the velocity in miles per hour or in metres per second, and S the surface in square feet or in square metres. The normal on oblique surfaces, at various angles of incidence, is given by the formula P＝KV²Sη, which latter factor is given both for planes and for arched surfaces in the subjoined table:—

Percentages of Air Pressure at Various Angles of Incidence

Planes (Duchemin Formula, verified by Langley). N＝P 2sinα/1+sin2α.					Wings (Lilienthal). Concavity 1 in 12.
Angle. α.	Normal. η.	Lift. ηcosα.	Drift. ηsinα.	Normal. η	Lift. ηcosα.	Drift. ηsinα.	Tangential  force. α.
−9°				0·0	0·0	0·0	+0·070
−8°				0·040	0·0396	−0·0055	+0·067
−7°				0·080	0·0741	−0·0097	+0·064
−6°				0·120	0·1193	−0·0125	+0·060
−5°				0·160	0·1594	−0·0139	+0·055
−4°				0·200	0·1995	−0·0139	+0·049
−3°				0·242	0·2416	−0·0126	+0·043
−2°				0·286	0·2858	−0·0100	+0·037
−1°				0·332	0·3318	−0·0058	+0·031
0°	0·0	0·0	0·0	0·381	0·3810	−0·0	+0·024
+1°	0·035	0·035	0·000611	0·434	0·434	+0·0075	+0·016
+2°	0·070	0·070	0·00244	0·489	0·489	+0·0170	+0·008
+3°	0·104	0·104	0·00543	0·546	0·545	+0·0285	0·0
+4°	0·139	0·139	0·0097	0·600	0·597	+0·0418	−0·007
+5°	0·174	0·173	0·0152	0·650	0·647	+0·0566	−0·014
+6°	0·207	0·206	0·0217	0·696	0·692	+0·0727	−0·021
+7°	0·240	0·238	0·0293	0·737	0·731	+0·0898	−0·028
+8°	0·273	0·270	0·0381	0·771	0·763	+0·1072	−0·035
+9°	0·305	0·300	0·0477	0·800	0·790	+0·1251	−0·042
10°	0·337	0·332	0·0585	0·825	0·812	+0·1432	−0·050
11°	0·369	0·362	0·0702	0·846	0·830	+0·1614	−0·058
12°	0·398	0·390	0·0828	0·864	0·845	+0·1803	−0·064
13°	0·431	0·419	0·0971	0·879	0·856	+0·1976	−0·070
14°	0·457	0·443	0·1155	0·891	0·864	+0·2156	−0·074
15°	0·486	0·468	0·1240	0·901	0·870	+0·2332	−0·076

The sustaining power, or “lift” which in horizontal flight must be equal to the weight, can be calculated by the formula L=KV²Sηcosα, or the factor may be taken direct from the table, in which the “lift” and the “drift” have been obtained by multiplying the normal η by the cosine and sine of the angle. The last column shows the tangential pressure on concave surfaces which O. Lilienthal found to possess a propelling component between 3° and 32° and therefore to be negative to the relative wind. Former modes of computation indicated angles of 10° to 15° as necessary for support with planes. These were prohibitory in consequence of the great “drift”; but the present data indicate that, with concave surfaces, angles of 2° to 5° will produce adequate “lift.” To compute the latter the angle at which the wings are to be set must first be assumed, and that of +3° will generally be found preferable. Then the required velocity is next to be computed by the formula

${\mbox{V}}={\sqrt {\frac {\mbox{L}}{{\mbox{KS}}\eta \cos {\alpha }}}};$

or for concave wings at +3°:

${\mbox{V}}={\sqrt {\frac {\mbox{W}}{0\!\cdot \!545{\mbox{KS}}}}}.$

Having thus determined the weight, the surface, the angle of incidence and the required speed for horizontal support, the next step is to calculate the power required. This is best accomplished by first obtaining the total resistances, which consist of the “drift” and of the head resistances due to the hull and framing. The latter are arrived at preferably by making a tabular statement showing all the spars and parts offering head resistance, and applying to each, the coefficient appropriate to its “master section,” as ascertained by experiment. Thus is obtained an “equivalent area” of resistance, which is to be multiplied by the wind pressure due to the speed. Care must be taken to resolve all the resistances at their proper angle of application, and to subtract or add the tangential force, which consists in the surface S, multiplied by the wind pressure, and by the factor in the table, which is, however, 0 for 3° and 32°, but positive or negative at other angles. When the aggregate resistances are known, the “thrust h.p.” required is obtained by multiplying the resistance by the speed, and then allowing for mechanical losses in the motor and propeller, which losses will generally be 50% of indicated h.p. Close approximations are obtained by the above method when applied to full-sized apparatus. The following example will make the process clearer. The weight to be carried by an apparatus was 189 ℔ on concave wings of 143·5 sq. ft. area, set at a positive angle of 3°. There were in addition rear wings of 29·5 sq. ft., set at a negative angle of 3°; hence, L＝189＝0·005 ✕ V² ✕ 143·5 ✕ 0·545.

Whence ${\mbox{V}}={\sqrt {\frac {189}{0\!\cdot \!005\times 143\!\cdot \!5\times 0\!\cdot \!545}}}=$22 miles per hour,

at which the air pressure would be 2·42 ℔ per sq. ft. The area of spars and man was 17·86 sq. ft., reduced by various coefficients to an “equivalent surface” of 11·70 sq. ft., so that the resistances were:—

Drift front wings, 143·5 ✕ 0·0285 ✕ 2·42	＝9·90 ℔
Drift„ rear wings, 29·5 × (0·043−0·242 ✕ 0·0523) ✕ 2·42	＝2·17 lb„
Tangential force at 3°	＝0·00 ℔„
Head resistance, 11·70 ✕ 2·42	＝28·31 ℔„
Total resistance	＝40·38 ℔

Speed 22 miles per hour. Power＝40·38 ✕ 22/375＝2·36 h.p. for the “thrust” or 4·72 h.p. for the motor. The weight being 189 ℔, and the resistance 40·38 ℔, the gliding angle of descent was 40·38/189＝tangent of 12°, which was verified by many experiments.

The following expressions will be found useful in computing such projects, with the aid of the table above given:—

1. Wind force, F＝KV2.	8. Drift, D＝KSV2ηsinα.
2. Pressure, P＝KV2S.	9. Head area E, get an equivalent.
3. Velocity, V=${\sqrt {\frac {\mbox{W}}{{\mbox{KS}}\eta \cos {\alpha }}}}$	10. Head resistance, H＝EF.
	11. Tangential force, T＝Pα.
4. Surface S varies as 1/V2.	12. Resistance, R＝D+H±T.
5. Normal, N=KSV2η.	13. Ft. ℔, M＝RV.
6. Lift, L＝KSV2ηcosα.	14. Thrust, h.p.,＝RV/factor.
7. Weight, W＝L＝Ncosα.

Aerostation.—Possibly the flying dove of Archytas of Tarentum is the earliest suggestion of true aerostation. According to Aulus Gellius (Noctes Atticae) it was a “model of a dove or pigeon formed in wood and so contrived as by a certain mechanical art and power to fly: so nicely was it balanced by weights and put in motion by hidden and enclosed air.” This “hidden and enclosed air” may conceivably represent an anticipation of the hot-air balloon, but it is at least as probable that the apparent flight of the dove was a mere mechanical trick depending on the use of fine wires or strings invisible to the spectators.

In the middle ages vague ideas appear of some ethereal substance so light that vessels containing it would remain suspended in the air. Roger Bacon (1214–1294) conceived of a large hollow globe made of very thin metal and filled with ethereal air or liquid fire, which would float on the atmosphere like a ship