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Aviation History
1927
1927 - 0896.PDF
70 TOFLIGHT THE AIRCRAFT ENGINEER NOVEMBER 24, 1927 accelerating thrust (that is the actual thrust, less the resistance due to the slipstream) is assumed to be proportional to the horse-power, that is :— accelerating thrust = T= X(HP) (1) This does not pretend to be exact, but is reasonably correct for the purpose we are using it. and for boats of the type under consideration. The water resistance of the hull is assumed to be of the form V Y-s = qw (— - ^ •(2) Where Rir is the water resistance w is the all up weight. q is a constant depending on the hull shape. V is the velocity. Vo is the velocity at " take-off." The above is equivalent to assumingthat the water resistance of a good hull may be taken as following a parabolic law with regard to velocity, the maximum resistance occurring at half " take-off " speed. This is not exact, but in Fig. 1 the assumed resistance is shown together with the actual resistance of a hull of normal English form. The air resistance is assumed to be eiven bv :— .(3) XdV dx gqV v'L The solution of (4) gives :— tan~ c .(5) where C = >AlX (q — I) — q- P Substituting V = V,, = the time to take off. K S'L in equation (5) we obtain 2Kv/PL (q-21)x'P C The solution of (4a) gives, where off," x = —I i logf i9(9 - ') L X tan x is the tan C run to -'(9- "take 2/)x'P (' 4000 •wjfln 2000 1000 0 (ft d> z £ Id a. / / / / ] 5 10 15 FULL SIZE 20 SPEE 3 IN ACTU <NOTJ W_RE£>ISTA^ Z5 30 35 RESISTANCE OIVEN BY / f y w i ) / Rw=qw[v0 yo^j q - -773 V \ \ \ \ \ \ \ \ \\ \ 40 45 50 Fig. 1. Comparison between actual and calculated resistance of flying boat hulls. where P.a is the air resistance I is a constant depending on the efficiency of the aerodynamic structure. Using these results, the equation of motion during tnke-off becomes :— or df- dS grX (H.P.) V,2 '• V.. w VdV dx yX (H.P.) w and PIf now we substitute KN'L for V where P is the power loading, L the wing loading and K is 1 ~F~i;~==, we getVpK Lmax. dV _g. ~dt = 1 KJL A simple way of finding .r- if / is cal<'ulated tirst is by the equation. P/ X K.'L fKN7ix = ^ < —-- loir, 2(q - I) \ g 1 r ql From (6) it can be seen at once the limit in "' take off "' (when the time becomes infinite) is given by :— (J = () that is when p _ 4X (q - I) This brings out the fact that with the assumption made W the limiting load a flying-boat will take off with is independent- pjpjT of the wing loading. This final load is, however, of no prac- tical value, what concerns us is the load with which a flying boat will " take off " with in a time of, say, two minutes, for this will be approximately the maximum time allowable. In equation (6) it will be noticed that as the time approaches infinity the value of the part within brackets approaches TT .(4) and never exceeds it. its value when the time is limited to approximately 120 seconds, we will assume to be 2-8 which is not far removed from the exact figures for a series of actual 8106
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