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Aviation History
1929
1929 - 0630.PDF
SUPPLEMENT TO FLIGHT 18 THE AIRCRAFT ENGINEER MABOH 28, 1929 FAMILY OF CURVES OF FORM V - kx n expression y — kx^ may be made to fit any plan form with a fair degree of accuracy. Fig. 1 shows a family of curves plotted for various values of n, together with pictures of the actual wing shapes corresponding with these plan form expressions. To facilitate construction, it is usual to make the leading and trailing edges straight and to fit circular arcs tangential to these edges to form the rounded tips. Fig. 2 shows the degree of approximation to this particular plan form that the parabolic curve will attain. There is no reason why the tip shape should not be of approximately parabolic form, blending into the leading and trailing edges at points A and B instead of being defined by the circular arcs R and r. In this case a structure designed according to the assumption that the plan form is as shown dotted in Fig. 2 would be on the safe side as regards strength. The next question to be determined is whether it will be satisfactory to assume that the load distribution along the wing is proportional to the chord. An answer to this may be found by referring to X.A.C.A. report No. 150, which gives the results of pressure distribution tests on a series of wing models, starting with a rectangular plan form and progressively tapering to a triangular plan. In this series of tests the wing thickness to chord ratio was made constant. Particulars taken from this report are shown on Fig. 3, which illustrates the effect of taper on the lateral position of the centre of pressure. It will be seen that when the taper is such that the ratio of the maximum chord to the mean chord is about 1-4 the lateral position of the centre of pressure is almost identical with that of the centre of area of the wing. It is interesting to note that most cantilever wings are designed with about this amount of taper. For the purpose of the following investigation a parabolic plan form denned by the curve y — kxi will be taken, the value of /,• for a span of 10 units and a maximum chord of 22 units being -- •= 1-17. The area of a wing of such a plan Da form is three-quarters of the area of the circumscribing rect- angle. It is easily seen that the ratio of maximum to mean chord is 1 • 333 and that the centre of area of each hajf of the wing lies at a lateral distance of 0-429 of the semi-span from the fore-and-aft centre line of the whig. This point is shown plotted on Fig. 3 and falls almost exactly on the average centre of pressure position found by experiment. The aspect ratio of a wing with these proportions is 6.1. Let us now consider the load applied normal to the wing chord on one side of the wing. W = total weight of the aircraft less the wing weight, i.e., the central load; A = total wing area ; N = load factor ; NWL = loading per unit area = ——; A. I = semispan. The plan form of the wing is defined by the curve y = kxi shown in Fig. 4. The values of W, A, and L are in accordance with the size of the aircraft and the desired wing loading. The value of k is found from the condition that when 2 2 I x = I, y = - 1, so that k = -. 6 (1) I may^now be found by integrating the plan form curve A between the limits of o to I and equating to —. 2* k I xldx = -Jo 2 I = 5A J (2) As we assume that the load per unit length along the wing span is proportional to the chord, the loading curve will be of similar shape to the plan form curve and is expressed by the equation :— y - L kx\ (3) PLAN FORM DEFINED FIG 2 10 -9 •8 -7 -6 PROPORTION •5 -4 -3 OF SEMI SPAN CAT O*—° —LATERAL POSITION Of CENTRE. OF PRESSURE^ ~g> FIG.3. 2586
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