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34 AIRCRAFT ENGINEERING February, 1932 A Discussion on Mathematical Lines of Certain Peculiarities and Disadvantages By J. Morris, B.A., A.F.R.Ae.S. APPARENTL Y two-bladed airscrews are acro- Th e angular velocities of the system of axes G A, dynamically more efficient in traction than G B, G C are respectively multi-bladed varieties, and, moreover, are ω =ω cos Ωt, ω = — ω si n Ωt, ω = Ω 1 2 3 easie r to manufacture. Furthermore, they arc Th e angular momenta of the airscrew about the simple r to transport. There were, however, axe s G A, G B, G C respectively are decide d drawbacks peculiar to the two-bladed h =A ω cos Ωt, h = — Bω sin Ωt, h = CΩ A n c airscrew . The more important of these are here an d about the axes G x, G y, G z discussed . h =A ω cos2 Ωt + Bω sin 2 Ωt h =(A – B) ω sin Ωt cos Ωt Th e Aerodynamic Couples h =C Ω An aerodynamic disadvantage manifests itself Th e rate of change of angular momenta about in a considerable periodic couple which is set up axe s fixed in space with which the axes G x, G y, when the axis of the airscrew is inclined to the G z are momentarily coincident at time t are in relativ e wind, as is the case, for instance, when engineer's units. a n aircraft is climbing,* This couple acts about a line perpendicular to the axis of revolution of th e airscrew and which line rotates at twice the airscrew speed. Th e author of this article is well Mr . Constant's Calculations know n for his mathematical investi Constant , in the course of his lecture on aircraft gation s into the theory of torsional an d W, B, B, C. If A>B, the whirling speed vibration, † said: "On the climb, on the turn, for W, A, A, C will be th e lower. vibration . He here , first, briefly sum an d in certain manœuvres, bad vibration and These whirling speeds may be far apart; for sometime s very severe vibration may be set up." marise s Mr. Constant's remarks on instance , the lower may occur at 10,000 r.p.m. In aircraft with a geared engine he calculates airscre w aerodynamic couples in a an d the upper a t 20,000 r.p.m. tha t couples of 6,000 lb./in. may be set up in Of course, the whirling speeds are dependent climbing , and of 10,000 lb./in . when turning, recen t R.Ae.S. lecture. He then pro no t only on the characteristics of the airscrew, but pulling out of a dive, or side-slipping. ceed s to give the mathematic s of the on the elastic properties of th e airscrew shaft. f t is important to notice that these couples will theor y of gyroscopic couples of two- b e absent in the case of a multi-bladed airscrew. Chance s of Failure blade d airscrews, which have not, Now, whirling speeds arise from lack of balance, Th e Dynamic w e believe, previously been published. whethe r static or dynamic, but between the two A dynamical disadvantage develops when the Finally , he recalls the whirling of whirling speeds of a two-bladed airscrew, how axi s of revolution of a two-bladed airscrew is turned ever perfectly balanced, vibration is impossible shaft s carrying two-bladed airscrews. uniforml y about an axis in the plane of rotation. an d either the airscrew or its shaft will apparently Th e article, therefore, constitutes a I n such circumstances, which may occur in a turn fail. This exponential instability appears to be and/o r a loop, a periodic gyroscopic couple is set due to gyroscopic action overcoming the clastic complete , thoug h short , accoun t of the u p over and above the ordinary steady gyroscopic resistance of the airscrew shaft in the speed peculiaritie s of two-bladed airscrews torque . To illustrate this, referring to Fig. 1, let rang e of instability. x G y be the plane of rotation of a two-bladed As B approaches A, the speed range of instability airscre w of principal moments of inertia A, B, C diminishes until, when B=A, there is no speed where , say, C>A>B. Let G be the centre of of instability if the airscrew is balanced. If, how gravit y of the airscrew and let G x be the axis ever, there is the slightest disparity between A abou t which the aircraft is turning. G y is per an d B, there will be instability at the whirling pendicula r to G x, and G z, the airscrew axis is speed.‡ norma l to the plane Ω G y. At time t let the angle I n practice it may never be possible to attain x G A = Ω where Ω is the constant angular perfect balance and perfect parity between A velocit y of th e airscrew about its axis of revolution. an d B. Indeed, the low tolerances allowed in G A, G B, G C represent the three principal axes aircraft airscrew manufacture may give rise, as of the airscrew at time t. Let ω be the constant Constan t points out, to considerable periodic forces angula r velocity of the plane x G y about the an d couples. axi s G x. Then the angular velocities of the system of axes G x, G y, G z are respectively g being the acceleration due to gravity. ‡ For a full discussion of this problem, see the author's "Strength 0 =ω , 0 = o, 0 = o 1 2 3 of Shafts in Vibration" (Crosby Lockwood, 1929, 30s.). Th e Gyroscopic Couples * See Aero. Res. Com., K. & M. No. 642, by H. Glauert. Th e corresponding gyroscopic couples set up on STANDARD HARDNESS NUMBERS † "Aircraft Vibration," by H. Constant. A paper read before th e aircraft will be a s illustrated in Fig. 2. It will the Royal Aeronautical Society on November 19, 1931. T o meet the need for recognised tables of be observed that the periodic couples will vanish hardnes s numbers for diamond pyramid indenta if B=A, which will be the case for multi-bladed tion tests corresponding to the British Standard airscrews. Brinell numbers, the British Standards Institution Fo r an aircraft fitted with a metal airscrew has issued B. S. Repor t No. 427, 1931, "Diamond of 11 ft. 6 in. diameter , an d revolving a t 1,000 r.p.m., Pyrami d Hardness Numbers." th e periodic couple of amplitude (a — b) Ωω may "Details as to the method of carrying out the test be 7,500 lb./in. when the rate of turning of the ar e laid down and tables of hardness numbers aircraft is 0∙3 rads./sec. With a wooden airscrew ar e given for loads of 5, 10, 20, 30, 50, 100, and of the same diameter, the couple will be between 120 kg. a quarter and half as great. A pyramid having an angle of 136 deg. has been specified as the standard, but an indication is Tw o Critical Speeds also given of th e allowance to be mad e for machines using an angle of 140 deg. Th e two-bladed airscrew also behaves in a Th e standard tables will be of direct interest remarkabl e manner from the point of view of t o all those engaged in the mechanical testing of critical speeds. Whereas a multi-bladed airscrew har d materials. designate d by, say, W, A, A, C, where W is the Copies of the specification (No. 427, 1931) can weight an d A, A, C th e principal moments of inertia, be obtained from the British Standards Institution ha s a single whirling speed, a two-bladed airscrew (W, A, B, C) has two whirling speeds corresponding Publications Department, 28, Victoria Street, t o those of two multi-bladed airscrews W, A, A, C S.W.1, price 2s. 2d. post free.
Aircraft Engineering and Aerospace Technology – Emerald Publishing
Published: Feb 1, 1932
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