Aeroplane Construction and Operation

by John B. Rathbun

Fig. 13. Flow About, Inclined Plane and Forces Produced by Stream. Fig. 14. Normal Plane with C.P. at center of Plate. Fig. 15.. C.P. Moves Toward Entering Edge When Plate Is Inclined to Wind.

A line OR is the resultant of the lift and drag forces L and D, this resultant being the force necessary to balance the two forces L-D. It is on the point of application O that the plate balances, and this point is sometimes known as the "Center of pressure." The center of pressure is therefore the point at which the resultant intersects the surface of the flat plate. The resultant OR is approximately at right angles to the surface at small incident angles, and the point O is nearer the front or "leading edge" (A) of the plane. The smaller the angle of incidence the nearer will the point O approach the leading edge A. By drawing OL to scale, representing the lift, and OD to scale representing the drag, we can find the resultant OR by drawing LR parallel to the drag OD and DR parallel to the lift line OL. All lines drawn through the intersection of LR and DR will give the resultant OR to scale. All of the lines must be started from the center of pressure at O.

The least resultant will, of course, occur when the plane is parallel to the air stream. The maximum resultant will occur when the angle of incidence is about 40 degrees, and on a further increase in the angle, the value of the resultant will gradually decrease. When the plane is parallel with the stream, the resultant is parallel to the plate, but rapidly approaches a position at right angles at about an incidence of 6 to 10 degrees. Beyond 10 degrees incidence the angle of the resultant increases past the normal.

The center of pressure (O), or the point where the resultant force intersects the plane, moves forward as the angle of incidence is decreased from 90°. When at right angles to the air current, the center of pressure is exactly in the center of the plane as shown by Fig. 14. In this case the drag (D) is the resultant, and acting in the center, exactly balances the air forces. In Fig. 15 the angle of incidence is reduced, consequently the center of pressure moves nearer the leading edge (A). As the angle continues to decrease, the C. P. moves still further forward until it lies directly on the front edge when the plate becomes parallel with the air stream. The center of pressure movement is due to the fact that more and more work is done by the front part of the surface as the angle is decreased. Consequently the point of support, or C. P., must move forward to come under the load. It should be understood that the plane will balance about the C. P. if a knife edge bearing were applied as at R in Fig. 15.

Calculation of Inclined Planes. We will now consider the inclined plane as a lifting surface for an aeroplane, and make the elementary calculations for such purpose. The lift will first be calculated for the support of the given load, at the given velocity, and then the drag. For several reasons, that will afterwards be explained, the flat plate or plane is not used for the main lifting surfaces, but the experience gained in computing the plate will be of great assistance when we start calculating actual wings.

Lift and Drag Co-efficients. The lift component (L) of the inclined flat plate depends on the velocity, area, aspect ratio and angle of incidence. Instead of using the co-efficient (K) formerly used for the total drag, we will use the lift co-efficient Ky. The formula for lift now becomes: L = KyAV² where A = area in square feet, and V = velocity in miles per hour. The lift co-efficient Ky, depends upon the angle of incidence. The horizontal drag D will be calculated from the drag co-efficient Kx, which is used in the same way as the co-efficient K in the case of the normal plate. The subscript (x) is used to distinguish it from the lift co-efficient. Both Ky and Kx must be corrected for aspect ratio. The drag can be calculated from the formula: D = KxAV² where the letters A and V are the same as above.

For the calculation of the drag, we will use a new expression—the "Lift-Drag Ratio"—or as more commonly given, "L/D." This shows the relation between the lift and drag, so that by knowing the lift and the ratio for any particular case, we can compute the drag without the necessity of going through the tedious calculation D = KxAV². The lift-drag ratio for a flat plate varies with the angle of incidence, and the aspect ratio, and hence a separate value must be used for every inclination and change in aspect. To obtain the drag, divide the lift by the lift-drag ratio. Hence if the lift is 1200 pounds, and the ratio equals 6.00, the drag will be: 1200/6=200 pounds, or in other words, the lift is 6 times the drag force. Changing the angle of incidence through angles ranging from 1 degree to 7 degrees, the lift-drag ratio of a flat plate will vary from 1.5 to 7.5. When the plane is parallel to the wind stream and gives no lift, the drag is computed from Zahm's skin friction formula.

The following tables give the values of Ky, Kx, L/D, and center of pressure movement for flat plates of various aspect ratios. The center of pressure (C. P.) for each angle is given as a decimal fraction of its distance from the leading edge, in terms of the width or "Chord."