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Aviation History
1919
1919 - 0014.PDF
a limit to height to which an aeroplane can ascend, and the exact manner in which tins makes itself felt aerodynamically becomes clear from the foregoing simple analysis of perform ance. In the same manner it is proposed to consider how the horse-power expended in turning affects the performance. Suppose the machine (Fig. 4) has reached the con. ditions of steady horizontal flight with velocity V in a circle of radius R, the rudder and other controls having been set to the appropriate positions to maintain the steady turning. The angle of bank being <p the equations of motion become :— Lsin <p = W/g. V«/R (1) L cos <p = W (2 D - T (3) terms of the horse-power required to drive the machine at the equivalent velocity of horizontal flight V, = V cos<p, - V/(i + V'/^R2)1". (10) no A. V W neglecting the relatively small components of the force on the rudder and the difference in lift and drag on the two wings due to the fact that the outer wing is moving more rapidly than the inner. For a given speed and flight path the requisite angle of bank is immediately given by tan <p = V*/gR. It follows at once that to fly in a circle of small radius with a given velocity the angle of bank must be steep, but equation (2) indicates at the same time that in order that the machine will not drop the lift must be imme diately augmented, demanding an increased angle of attack compared with horizontal flight. This involves a greater expenditure of horse-power. In general performance curves for any machine usually give a measure of the horse-power required to maintain horizontal flight at various speeds. From such a curve the corresponding horse-power required to maintain circular flight may easily be derived. In either case the horse-power is expended in overcoming the drag. Comparing the two cases, horizontal flight at velocity V and circular flight at the same speed but necessarily at a different angle of attack, H, - T,V - D,V - Ap(D,),V» (4) for circular flight, Hs - T2V = D3V - Ap(Dv)2V» (5) for horizontal flight, where (Dr), and (D,-), are the appropriate drag coefficients. Therefore H, - H2(D,),/(D,)2. (6) Now W = Ap(Lt),V2 cos <p from equation (2), and W - Ap(Lt)2Va for horizontal flight. Therefore (L<)./(L<-)i= i/cos<p. (7) Suppose the machine flying in straight line flight with velocity V, at the angle of attack corresponding to the lift coefficient (Lr)„ that of the previous circular flight, then W = Ap(L,)iV,» and W - Ap(L,)3V«, as before, where V is the horizontal flight equivalent, from this point of view to the circular flight. Therefore (L,),/(L,), - V'/V,' hence V, — V N/cos <p (8) If H', and H* be the horse-powers required to drive the machine in horizontal flight at velocities V, and V respectively then H., = DSV = Ap(D,)2V», and H'^D.V, = Ap(D,),V,», therefore (D,),/(D,), - H',/H,. Vs/V,» -H',/H,. i/(cos<p)»'2. This gives finally, using (6), H, -HV(cos<p)»'2 (9) expressing the horse-power required to maintain the machine in horizontal circular flight, of radius R and velocity V, in The method of obtaining the horse-power curve for circular flight is at once obvious from these formulae. The horse power for horizontal flight at velocity V ^/cos ^> is divided by (cosip*'s) and plotted on a velocity base at abscissa V. In Fig. 5 AD represents the horse-power available from the propeller of a particular machine, and CB the curve giving the horse-power required at various speeds for horizontal flight. The curves of the type PQ representing the horse power for circular flight at various radii and speeds are obtained by the method indicated above. Since SM represents the horse-power used up during the flight, LS measures the surplus horse-power available, say, for climb. The curve of the type PSQ, passing through L, will, except in the cases about to be mentioned, fix the minimum radius of the circle at the velocity determined by M. If CN is the locus of the points on each curve corresponding to maximum .lift coefficient, none of the curves of the system can possibly go beyond CN, and the points of intersection of this system with this line determine the maximum radii of circular flight at the corresponding speeds within the range limited by C and N. The minimum possible radius corresponds to the curve touching AB, in this case approximately 180 ft. It must be remembered that in the type of flight con templated it is necessary that the angle of attack of the \so X Q IUU SO 0 ! 1 • - A^ 1 fc?^ 1 lAg2^~- .' _ LS± . 1 ... J / / \/ A yS \ JS * -^>^l / > /V V n< ^ * p. n> (0 M 'o« '10 110 IK, IM iso its ifo ito 100 5PECD (FT/5KJ machine should correspond to a higher lift coefficient than that of horizontal flight at the same speed, in order to sustain the weight when banked. The corresponding drag coeffi cient would then be greater or less than that of horizontal flight, according to whether the point considered lies above or below the position of minimum drag, excluding those cases '4
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