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Aviation History
1912
1912 - 0124.PDF
I/Hii FEBRUARY IO, 1912. LONGITUDINAL STABILITY. By 0. D. ATKINSON. MR. MERVYN O'GORMAN, in the paper which he read before the Incorporated Institution of Automobile Engineers last year, touched on the problem of longitudinal stability at some length, but although he showed some interesting graphs, he did not give the mathematical proofs of his statements. It is the purpose of this article to give these latter. Mr. O'Gorman's conclusions on this s lbject were two in number, namely, that for longitudinal stability t lie angle of incidence of the front plane must be greater than that of the rear, and that the so-called " tail-first " type is more stable than the other two types, namely, the Farman and the Bleriot. Let us imagine an aeroplane, the surfaces of the front and back planes being Sj, S2 respectively, and their angles of incidence a, 0. & ODATKtNSOH. I^et the distance between their two centres of pressure be x, and let t' be the resultant centre of pressure. The longitudinal stability of this aeroplane owes its existence to a great extent to a righting couple, whose moment varies as the travel of P. Let the aeroplane be tilted up through an angle 0, and let us take the first case, when o = /8. Then, before tilting, P divides x in the ratio of ;>—',.„ T—==?» Kb3 Vz sin /8 S2 not the lift, being employed. the formula for the total resistance, After tilting, P divides x in the ratio of KS,V 2sin(a + 0) §1 s2-KS2V2 sin 08 + 0)" . •. P does not travel, and the aeroplane is in neutral equilibrium. Let us then take the case where £ > o. In normal flight, P again divides x in the ratio of ^—.—. S2 sm /8 S, sin(a + 0) When the machine is tilted through 9 this becomes S 2sin(0 + 0)' and P travels forwards or backwards, according as to whether S, sin(a + 0) S 2sin(i8 + 0) , S 2 sin a is greater or less than =*-;—-. 0 S 2 sin /3 But since fi > a, cos (0 - a - 0) > cos (a - 0 - 0), .*. Cos(/B-a-0) - cos(a + /8 + 0)>cos(a-/8-0) - cos(a + 0 + e), .". sin0sin(a + 0)>sinasin(/8 + 0), sin (« + ») sin a *'• sin(/8 + e)>sm~^- .'. P travels forward and the aeroplane is in unstable equilibrium. If., however, a > $, the reverse of this occurs, and stable equilibrium is obtained. By tilting the aeroplane down instead of up, we get similar results to those above. All through this article the travel of the centre of pressure on each plane due to the increase of the angle of incidence has been neglected, as this is very small compared with the total travel. Mr. O'Gorman's next point was the relative sizes of the front and tail plane. Let KS2=SH and let the travel of the centre of pressure be /. Then t=xl KS2sina KS2sin(q + fl) \ lKS2 sin o + Si sin 0 KS2 sin (o + 0) + S2 sin (/3 + 8) ' Let sin a, sin£, sin(a + 0), sin(/8 + 0), and —equal a, 6, c, d,y, respectively. Then y = «K cK K(ad- be) acK? + K(ad+bc) + bd aK + b cYL + d AK BK2 + CK + D' _ A(BK2+ CK + D)-AK(zBK + C) lheadK- (BK^CK + Df ^/CZ-4BD-2BC Let this=5 dy When dy = 0, K rI> 2B AD - ABK2 (BK2 + CK + D)2" the former and sin o > sin /3. dK "' " V B gives the maximum positive value of y. /b~J_ /sin g sin (fl + 9) V ac~~ V sinosin a + 0)' .'. K is a fraction, and Sj is smaller in magnitude than S2. Whea the tilt is downwards, K, by similar working, becomes equal to V slngsin(fl-e) sin a sin(a-0) If we add a front elevator to a Bleiiot type we have a Farman type. In practice the angle of incidence of the elevator is generally 0° in normal flight, and it is easy to see, without wading through more heavy working, that as the tilt increases, the elevator exerts a lift bringing the centre of pressure forward, and vice versa, thus decreasing the stability of the machine. If a flat tail plane is put on to a " tail-first" type, however, exactly the opposite occurs, and the stability is increased. At first sight it may seem that the value of K has solved the problem of stability, but a more careful reading will betray that it contains 9, a variable ; still, if it does nothing else, it serves to show the superior inherent stability of the "tail-first" aeroplane to the other two types. ® ® ® ® University College Lectures. AN introductory lecture on the theory of the aeroplane was given on February 2nd at the University College, Gower Street, by Mr. A. R. Low, who will begin a series of class lectures on February 9th. When these are completed, the College will hold an examination and issue certificates. Territorials at the Royal Aero Club's Eastchurch flying grounds practising with "Short" biplane, No. 32. 124
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