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Aviation History
1917
1917 - 0212.PDF
MARCH I, 1917. THE AEROPLANE OF TO-MORROW. By LOUIS DE BAZILLAC, Engineer (Ecole Superieure d'Aeronautique de Paris). Translated by B. BRUCE-WALKER, B.Sc. Continued from page 190.) APPENDIX II. The maximum of each of the climbing speeds N is attained or a value of S smaller than ["-t-^wA^ where A is the detrimental resistance to forward motion ; wthe weight of a square metre of lifting surface; and W/, the weight of the machine without wings at sea level. It has been proved above (articles of September, 1914) without giving the corresponding values of S, that the vertical climbing speeds Nu, No, and N\ expressed as functions of S, pass through a maximum. It can be shown, to complete tHe caldjation, that the maximum of each of the speeds N is attaijB for a value of S generally smaller than a certain well-defflpd magnitude: w Kx For this it is proposed^to determine the maximum altitude attainable, as a function of the surface area, and the times necessary to climb to different altitudes. Let Z be the altitude of. the machine above sea-level, V its velocity at this instant, and V* its velocity at sea-level. To simplify the calculations, we will impose the condition : V = V* 7 • W putting 7 -= h being the atmospheric pressure at the point considered. We know also that Z - 18,400 x 2 log 7; or again Z = 2 A log€ 7; putting A = 8,000 The relation, V - V* 7 immediately gives the law of the variation of the speed as a function of the time, and of the direction of the tangent of the trajectory. Lastly, the vertical speed is = V sin m ;d f u> being the angle of the tangent to the trajectory with ihe horizontal. Z •= 2 A loge 7 = 2 A loge — whence dZ df 2 A ^-• • = V sin » and sinu ( 2 A Notice that the condition V = V6 7 has the result of rendering the resistance of the air independent of the altitude. In fact, the lift and the drift are given respectively by: and R - R* R* V*3. (3) y This granted, the equations of the motion are obtained by the successive projection on the tangent and on the normal to the trajectory of the forces applied to the system : H - R - W sin w = m ~ ; dwF - W cos m - m V —-.. V a denoting the tractive force of the propeller ; R the resistance of the aeroplane to forward motion - R* V**; F the thrust normal to the trajectory - Ry Vt:', W the weight of the machine.w its mass = •-•; V its velocity at the instant considered = V/, 7; tt> the angle of the tangent to the trajectory with horizontal. To these two equations it is necessary to attach the con- dition : V* • dV ^X«aw-Tf Let us now write the equation of the tractive force of the propeller. HU, at the level of the ground. This equation can be written with sufficient approximation: H* =i-«Y2 (4) the where the coefficient a is generally in the neighbourhood Of O'll. Again, the tractive force H in an atmosphere at the pres- sure h is equal to: H, -J? H - H* -V= 760' = b V/, 2 The equations of motion may then be written : R, V** = b?£ - flV*« - W sin « (1 + (^ ),; R = W cos w + m V-Jdf (5) v .= —v smAdV, df 2A Having arrived at these results, we will suppose that during the climb the variations of the angle of attack and of the surface area are slight enough to allow the supposition that Ri remains constant. This hypothesis, supported, moreover, by experience, for the variations of angle of attack permits the setting aside of the second equation of the motion, the first immediately giving the equation of the hodograph : - (R* + (6) This equation being no longer a differential equation, it is permissable to neglect the term as compared to unity neighbourhood of 0-03V 2 is in the We take, then, as the equation of the hodograph :— Ri- and Ry corresponding to the limit of the angle of attack at the summit of the trajectory. Equation (jf, by introducing the variable 7 =» _.- and patting p — ~-^—> can again be written:— sin Vs (8) Where ji is the value of 7 corresponding to the maximum altitude (summit of the trajectory). The time necessary for climbing to a certain altitude will be given by equation (2). • = P (^ - 1) JVt V- sin * = A . J. . Log.zH^r . yi^J. (9) p V* 7 7i — 7 7i + I Suppose, that without altering the propulsive mechanismor the general design of the wings, the surface S is increased. Let:w be the weight of a square metre of lifting surface; W4 the weight of the machine without wings ;X the detrimental resistance to forward motion. When the machine has reached the summit of the trajectoryit can be assumed that it satisfies the conditions of horizontal flight, or that then rf* = O.df dV dV V2Besides this *,->••= O in the limit, since —, = -- sin w. df df 2A We have, then, at the summit of the trajectory, by replacing V by Yt, Y, equation (3) :— 7* whence W 72 0 Ry KyS- 2 i/ 7l~ ~" w7+~wS • K*S + (a + an expression which increases with S so long as S < I z-i-y: m Wl 212
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