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Aviation History
1936
1936 - 1109.PDF
SUPPLEMENT TO FLIGHT 4626 24 APRIL 30, 1935 THE AIRCRAFT ENGINEER span will then be taken in the form of a Fourier Series of the usual type, viz. : A, sin 0 + A3sin 30 -f . . . . A2n_, sin (2« — i)0 -f- . '•. (3) the coefficients A2„_, being evaluated by such frequently employed methods as those of Glauert or Lotz. • < The semi-span will be considered to be of unit length. (i) Lateral Co-ordinate (Y) of M.A.C. The fundamental formula, on replacing KN by KL, may be written : where : XT = ordinate of the locus at the wing tip. a '==, length of parallel centre section. The fundamental formula then becomes : — r° • , xT-a r* 3 A-xTa X a \ KLc[c0 dy + • KL cjcuy dy f I — a I —a fl KL <•;<•(, V dy KL c/c0 dy <* • • • (4) KL c!cQdy (10) Substitute, as before, the series (3) and let 0 = 6„ for y = o. Evaluating "each of the integrals in (10), in turn, we have : Substitute the series (3) in numerator and denominator, at the same time altering the independent variable to 0, where y = — cos 0. Hence as y varies from o to 1 over the positive semi-span, 0 will vary from ir/a to v. ri rir n KLc/c„r dy = —1/2 Z A,,.,_, sin (211 — 1) ir'2-l 11 — 1 8 . sin 2 0^0 r 0» n KL cjc0 dy -= A •n/2-J «=£ sin 2 0o w/2 /sin 2 (« — 1) 0O sin in 0O 27 A2n_, sin (2W-1) 0.sin O.dl + 1/4 2 A2,:_, ?; = 2 (« - 1) 2" 0»\ n J 11) = 1/4 2: A, »= 1 sin (2» — 3)77/2 sin (2>i -f- i)w/2 ri (2" - 3) (2« + I) KL c/r0 dy = fir 7? (5) Also : ~Al[ * A2](_, sin (2n — i)0 . sin 0 . rftf sin 2 0„ KL c/c0 <fy = nA, •TT n Z A2n_! sin (2ti — 1) 0 . sin 0tf*0 ir/2J ?? = I ••)-i/4 i7 A,M ' n = 2 sin 2 (» — 1) 0O sin 2« 0OS (w — 1) n (6) • • • • Hence Y~ = n Z A 2 „ _ j w=i sin(2w — 3)77/2 sin(2w 4- i)*/2 0- 77A K L c/c0 rfy =* —^ by (6) 4 (12) (13) (2« - 3) (2W -(- I) 77 A, KL c/c„y rfy = — 1/2 27 A2„_, . sin (zn - 1) —3 27 (- \r * 2" — 1 77 wl, (2H -3) . (2W + I) ' A, (7) 0. sin 20 . dd n 0J n = 1 0 ~i/4 27 Af|t_i(-»i"(»"-3)^_smf2W + i)1. This result is true for a wing of any plan form. Since in practice it is customary to employ only four terms of the series (3), under this condition we have n=x (2w - 3) (2W 4- I) 1 A3 1 A5 1 A," 1/3 "\5 A', »A, ! 45 A J •• •• (8) Note ; For an elliptical load distribution, the coefficients A3, Afc, A7, etc., are each zero and we have the result: Substituting for (11), (12), (13) and (14) in (10) we then get 1 1 r A,/ sin 21 X = tfA, a-— 0O - w/2 -—-[_ I 2 \ 2 4 I \ " -°)+i/4 X A / ' W = 2 in-I Y = - = .425 3* /sin 2(w — i)0„ ^_ sin 2H0O* V « — 1) « / J (9) (ii) Longitudinal Co-ordinate (X) of M.A.C. The locus of the aerodynamic centres of the individual chords is indicated in Fig. 1. This locus will be assumed to intersect each chord at a constant fraction of its length. (The theoretical value* of this fraction is one-quarter, measured from the leading edge.) Thus over the rectangular portion, a, of the plan form, the locus will lie at a constant distance " a " from the lead ing edge, and over the trapezoidal will be given by the linear equation : Xx — a a — X-[o X = . y -f- v * -1/4 27 A3„-i « = 2 .. ('5) I — a l—o * Bv a theorem of N. E. Joukovski. In practice too this is the value employed. XT-« /sin (2n-3)0„_sin (swj^WAI 'TT^TeA1*^ 2n-\ (2—3)" a'+ X-rCOS 0O \Atf sin 2 0 H i 21 1 — 1 w — 0O -f 1 -f cos 0O [2 \ sin 2(w — i)0„ sin 2n0o\| (n - 1) n )\ which eventually reduces to the form : X = Aa + (1 - A)XX +(a - XTJ ( «I + f ""' -£p (16) . • * • where : 77 + (2 0O — 77 — sin 2 0O) cos 0n 7r(l -f- COS 0O)
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