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Aviation History
1940
1940 - 2424.PDF
Supplement to ™ AIRCRAFT^ ENGINEER No. 175. l5th August 29, 1940. THE SPEED CUBIC An Alternative Solution By RICHARD M. CLARKSON MR. KEITH TURNER'S interesting article on thesolution of the Speed Cubic in Flight prompts meto submit an alternative method of solution which may possibly interest your readers. It has been in use for over two years and has the advantage of being simple, quick and accurate ; it avoids the use of awkward roots, and keeps the physical significance of the quantities in view. We have HP, = HP,, + HP, OT where HP, HP,, (I) HP,, = H.P. available (thrust power). HP, = Parasite drag H.P. HP, = Induced drag H.P. Expressing the above quantities in terms ol speed, drag, weight, atmospheric density, airscrew efficiency, B.H.P., and span loading, we get :— D . a . V3 = i — t where K = Factor .for non-elliptic load distribution. W/S2 = Span loading. W/ryP = Thrust power loading. Equation (2) is solved by means of the two curves on the attached figure, which are self-explanatory. It is necessary to approximate a value for V in estimating r, but since, for conventional aeroplanes, r normally lies between 0.05 and 0.20 and the value of (1 — r)* is relatively insensitive to r. One approximation only is generally necessary. Naturally D may be expressed in any other convenient form by suitable adjustment to the constant in equation (2). It is interesting to note that curve A gives the speed that the aeroplane would do if there were no induced drag, t = o corresponding to no induced drag, r = .5 to flight at max. L/D where the induced drag is half the total drag. By differentiating Eqn. (2) with respect to each of the four variables in turn we obtain the following relations : d\ or i 77 X iol x v = 121 (^-) x (i - r) (2) in which V = True airspeed (M.P.H.). t; = Airsciew efficiency. P = B.H.P. D = Parasite drag ot aircraft in lb. at ioo ft./sec. o- = Relative atmospheric density. _ HP, _ 0.33 K x W/,P x W/S2 ' ~ HP: V^T* = *P (say) = «* V do 3 - 47 ~ w " which are plotted on the graph overleaf (Fig. 2). This is a useful companion to the other graph and shows the relative sensitivity of level speed to the several variables which determine it; and demonstrates further the im- V D V W V 0 d-qP dW dD ~ dV ofW ~ dV = ' 3 1 3 3 _ 1 - V — y - 4? 2r — 4* — 2t 5OO 4OO Vo M.RH 300 200 100 CURVES FOR SOLVING FOR V IN THE EQUATION:- V 0-rfs 1. OBTAIN Vo FROM CURVE A 2. OBTAIN (l-r)% FROM CURVE B 3. THEN V'VoO-rO'4 I I I I I I I I I I I I I I I I I I I I I I 1 I L I 1 I I •0-5 r>\ 1 1 1 1 1 TtP
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