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Aviation History
1932
1932 - 0985.PDF
SEPTEMBER 30, 1932 $7 THE AIRCRAFT ENGINEER SUPPLEMENT TO FLIGHT R = WP 10-00237 pSV2 6 + 3600 x 55(h) ( The cruising speed is taken as K X most economical cruising w»(<|> + C) ) 0-00237 pSV2 J 11) Endurance H hr. = speed, i.e., V = KVC = K. X S <i> + C x w i ,2 ,0-00237 p where K is any chosen constant. Substituting for V in equation (11) and simplifying, we get Range R ft. _ 3600 550TI K2 1 WP ~V ' (1 + K«) ' [6(4) + C)]* ' ~W i.e., Range 375TJ K2 1 = ~y ' (i + K*> ' Tfr^Tc) w ft. -^ miles . . (12) (b) Range at a constant proportion of the most economical cruising speed taking into account the variation of this speed with gross weight of aircraft. Let W0 = gross weight at beginning of flight WF = gross weight at end of flight. i.e., Weight of petrol used = W0 — WF. NofE.—Strictly, the amount of oil used should be taken into account in the weight change, but this is a very small correction, and may be neglected. Now in this case <*R = ^ m dR dW P I W2(A + C) i 3600 x 55CM,°-00237PSV29+ 0-00237p SV2f " (13) Taking, as before, V = KVC = KX^/: 6 "\0-00237 p and substituting in (13), we get by simplification 3600 x 5507] K2 1 dW d~R = p (1+K«) ^/6(4» + C) W Integrating between the limits W„ and WF gives 3600 X 550TJ K2 1 Range R ft. = W0 i.e., Range R = A log —- miles WF P (i + K4) v/e(<j, + o w x 2-3 log-/ft. WF (14) where A K-8637) P (1+K4) v/e(4> + C)' The similarity between this formula and the formula of Breguet will be noted. The endurance under these conditions is found as follows :— If dW = weight of petrol used in dK hours, «H = — where P = consumption in lb./hr. From equation (8) dH = dW P 'o-OOBTpBW W2(4> + C) • (15) 550T) and as V = KV„ = K x /^/* 0-00237 pSV | W — x (- 6 < \0-00237pS By substitution and simplification «fH = 55^> K x (0-00237 p 8)4 dW P (1+K4) v/e(< , + C)3 WV2 Integrating between the limits W0 and WF gives 550T; X K p (1 + K*) (0-00237 p S)i j" = B W.1 where B IIOOT] x 1 W0i K W04| VQ($ + c)3 \wFi hours (16) y/0-00237 pS p (1+K«) v/6 < > + C)3 Then the mean cruising speed during the flight is given by Range R = — = Vm m.p.h. Endurance H The expression for Vm m.p.h. is found to be '+ + CV V™ =0-784 (W0WF)i e (W0* - WF*) K log W„ WF m.p.h. (17) (0-00237pS)4 This is obtained by dividing Equation (14) by Equation (16). From the endurance formula (Equation 16) the value of K to give maximum endurance may be calculated easily by differentiating H with respect to K and equating to 0. •jr i.e., H = Q where Q = dR '"' dK~ .-. K = 1 +K4 11007] y/0-00237pS ( 1 ^ WF4 V Q(i 3K») Wo* I (1 + K4)2 = O for a maximum i.e., the speed for maximum endurance is 0-76 X most economical cruising speed. This can be shown also by differentiating P (the consump tion in lb./hr.) with respect to V and equating to zero, since the speed for maximum endurance is, obviously, the speed at which the consumption is a minimum. It should be noted that the speed for maximum endurance is not a practical flying speed, being very near the stalling speed of the aircraft and also outside the range of the assump tions made at the beginning of this article. To use the formulae, the most economical cruising speed Vc is first obtained from equation 10, using the aircraft characteristics, and then the appropriate range formula used for any desired speed KVc. These formulae, due to the assumptions made, do not provide a means of estimating accurately the range and endurance throughout the cruising range. This can only be done by using actual horse-powers, specific consumptions, and airscrew efficiencies. However, based as they are, on the fundamental characteristics of the aircraft and engine, they do give a rapid means of indicating the range at any cruising speed, sufficiently accurate for preliminary design work, before full performance estimates are available. APPLICATIONS OF THE POLAR DIAGRAM By E. H. ATKIN, B.Sc.(Lond.) (Continued from page 64) Strut No. 4. Middle Half Stiffened This strut is 2£ in. O/D 16 S.W.G. at the ends, 130 in. long and the middle 65 in. has a moment of inertia twice as great as the ends. It is subjected to an end load of 4,360 lb. The eccentricity is calculated as before, hence 130^ 2-5 ~ 600 40~ = 0-2795 in. 918 c
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