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Aviation History
1917
1917 - 0189.PDF
PKBRUAST 22, tically incompatible for A. zero and for S infinite, and that (A) negatives (B,). Equation (B,) Modified—The Formula proposed—ItsAdvantages. What must be done for system (BJ to remain in its form and general character the accurate expression of the problem ?Tan 7 must be a function of a,, so that for S = 00 or A =» o, 0^ = 0); we must have, in fact, if a is a coefficient relating to the body and the wings :— <r + roj2. r rKS and tan 7 = ra. -j- awhich gives, for S = 00 :— OQ = a!;and tan 7 = / -|- 2 m,, = t -(- 2 n^. Formulas (C,), generallv accepted, then become :—R* = KSV2 [ra2 + ta + (a + raf)'] Ry - KSV^aso that (BJ and (B 2), (C[) and (G>) are identical, provided wewrite :— s = <r -f- rax~.This interpretation of the value of s is more rational and more general, since it contains the case of a, not being zero,and since for a, zero s — tr. It allows us to consider now analytically as well as accord-ing to the data furnished by experience, an angle a u the bestangle of the wings, not necessarily zero, which it is possible to adopt for the normal angle without being led to give aninfinite value to the tractive force. Application.Before stating the consequent effects on the theory of the aeroplane brought about by the modifications = <r + roj2 a direct application of the formulae (C.) will be given.Suppose it is required to calculate for an aeroplane of variable surface in which the normal angle is the best angleof the wings :— (1) The minimum surface necessary for sustentation witha tractive force H ; (2) The surface giving the maximum vertical speed whenclimbing at the angle corresponding to this speed. Let :— G be the centre of gravity of the machine, W its weight, assumed constant,V the speed, * the angle of this velocity with the horizontal, •and H the tractive force of the propeller, assumed con- stant and directed parallel to the velocity of the machine. If we resolve the forces acting on the machine first alongthe direction of the velocity and then perpendicular to it (see Fig. I of the article of September 4th, 1914), we canwrite ia a first approximation :— H — W sin i = KSVa (ra" + ta + r + ra,2); Wcosi = KSV'2a;Whence dividing one equation by the other and taking for small angles the tangent equal to the angle and the cosineas unity:— H . . . . . <r + ra,2 W _ i = 0 _ i = ra •+• t + and the vertical component N of the velocity gives : In looking for the maximum of N it will be found that ittakes place at a value OM of the angle of attack defined by the equation : ro-M +(8-t) a,, - 3 (<r + m,2) = o of which only the positive root need be considered. Equation (2) gives for aM the value :—• - tf (2) (3) It is evident that aM is always higher than a,.We will now see that o M is always lower than a0. For thiswe will change equation (2) slightly. Notice first of all that for a = o,, N = o ; and equation (1) gives :— $ = 2r«, + t A— ; for a = a^, 6 is a minimum, its derivative vanishes and Substituting these values in (2) it becomes :•— «„ - (4) (S) (6) It is easy to see that the value of oM that satisfies thisequation is lower than a,,. For the values of o M equal to«i and OQ, the polynomial (3) passes, in fact, from negative to positive. The positive value of an that makes the poly-nomial zero is therefore necessarily placed between o, and OQ. Consequently :— o, < OM < a,,. For S = =0 the relations (3). .4) and (5) show us that o0 = a.M ass O!. These results are in every respect in accord with those furnished by the experimental formula :— tan 7 = Kx + g Ky in which, for S infinite, a,, is equal to n, and tan 7 to the value of the ordinate correspondiing to the minimum of Ky Formula (Bj) would have given 6 = 00 with : an = OM =5 o. The maximum value of N—which we will denote by N»i —can easily with the aid of equation (2) be put in the form (article of September nth, 1914) :— 2 r s/aM so long as the value of N corresponding to o0 is expressed by:— / w N,,= \/•£§ and the value of N; corresponding to a, by :— By studying the variations of N»t, Nu and N, as functions of S and taking their derivatives, we would obtain the curves given in Fig. 5 in the article in " FLIGHT " of September 1 ith, 1914. These curves all pass through a maximum and have zero values for S = 00. No and NM vanish for :— . \ . '' ' The speed N, vanishes for :— S Ka, [(6 - /) - »«,]" It will be seen that for 6 W a> these curves all start from the origin of the axes. For 0 — t •= 2rat each of themaxima lies at a point of zero ordinate for S = 00. For no surface area an aeroplane then needs an infinite tractive forceor no weight to sustain itself in a horizontal path. With an infinite surface area it suffices to have a tractive forceH = W (t + 2ra,). Formula (B^ has led us to say: "With an infinite surfacea tractive force H = Wt is needed," which is not a general truth, but is only true in the particular limiting case of Oi = O. Consequences of the Modification brought about by System (C2) in System (CO-Of the modification brought by system (C,) into system (d) by means of the expression :— s = tr + rat- the following consequences make themselves clear :— 189
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