In the engine-out example just presented, you may have noticed that the airplane was really not in trim. With the rudder deflected to balance the yawing moment of the engine, a side force is produced that must be counteracted by some means. This can be done by rolling the airplane around its x-axis to give a component of the weight along its у-axis. This is shown schematically in Figure 8.27. To carry the preceding engine-out example further, the sum of the forces along the у-axis must be zero for trim.
+W(f> = 0
Thus, to trim the engine-out situation completely, the pilot must have the capability to roll the airplane. This is accomplished by movable control surfaces that are hinged on the outer rear portion of each side of the wing, as shown in Figure 8.28. These surfaces, known as ailerons, move differentially: as one moves up, the other moves downward. Since the ailerons do not necessarily move the same amount, the aileron deflection is defined as the total angle between the two ailerons. For example, if the right aileron moves down 10° and the left aileron moves up 14°, the aileron deflection, Sa, is equal to 24°. Sa is defined to be positive when the right aileron rotates in a positive direction.
It was the incorporation of roll control that distinguished the efforts of the Wright Brothers from those of the other aviation pioneers of their day. Instead of ailerons, however, the Wright Brothers warped their box wing to provide a differential angle of attack from one wing tip to the other. This was improved on by Glenn Curtiss, who developed the hinged aileron, which was a considerable improvement over having to warp the wing. Since they had first recognized the need for complete lateral control and had devised the means to do so, the Wright Brothers brought a legal suit against Glenn Curtiss, claiming that his ailerons infringed on their patent rights. After a lengthy and bitter legal fight, the courts ruled in favor of the Wright Brothers. I have only the greatest respect and admiration for the Wright Brothers, but I question this particular legal decision, since it seems to me that Curtiss’ development of the separate movable aileron control surface was, indeed, a
control system distinct from the wing-warping system developed by the Wrights.
Leaving history and returning again to the technical aspect of roll control, observe that, when an airplane is rolled through an angle, ф, there is no mechanism present to generate an aerodynamic restoring moment. In the preceding cases involving angle of attack and sideslip angle, we encountered the derivatives CMa and CN/j, respectively, that had nonzero values. In the case of the roll angle, however, the corresponding derivative СІф is always zero. Because of this, the concept of roll stability, in the static sense, does not exist. At the most one could say that an airplane possesses neutral static stability in roll.
We define the rolling moment coefficient by reference to the wing area and wingspan, as with the yawing moment coefficient.
L = qSbC, (8.93)
Here we have a real problem with notation, since one tends to think of lift when the letters L or / are used. You will have to be careful and aware of the application of the coefficients in order to make a distinction. Generally, if the lowercase l is being used with reference to the entire airplane, it has reference to the rolling moment.
The rolling moment produced by the ailerons can be estimated by reference to Figure 8.29. As shown, the inboard end of the aileron is located a distance of у! from the wing centerline. The aileron extends from there out to a distance of y2. As a result of the aileron deflection, an increment in the section lift per unit span is produced on the right side and is given by
d(AL) = qca0T SaR dy
This incremental lift results in a differential rolling moment.
dL = – yqca0r SaR dy
Integrating this over the spanwise extent of the right aileron gives
L = – qa0T SaR I су dy ■Li
Since the wing is symmetrical, it is easily shown that a similar contribution to the rolling moment is obtained from the left side, so the total rolling moment is given by
In dimensionless form, С/, becomes
Ci = —a0rSaA j (j^)xdx
where x = yl(bl2).
[3(*22 ~ *i2) – 2(1 – A)023 – де,3)]
For a linearly tapered wing this can be integrated to give
This relationship for C( is only approximate, since the increment in the section Ci is assumed to be constant over the aileron. Induced effects associated with a finite lifting surface are neglected. By comparison with experimental measurements (Ref. 8.8), it is obvious that these induced effects are significant and cannot be neglected.
If the aspect ratio of the wing is fairly high, say approximately 6 or higher, then one can use a fairly simple model to correct Equation 8.96 for induced effects. Otherwise, a set of graphs, found in References 5.5 and 8.3, should be used.
When deflected, an outboard aileron will produce an incremental lift distribution something like that sketched in Figure 8.30a. Thus, on either end of the aileron, an incremental vortex system is shed in a manner similar to that shed at the tips of a wing. This incremental vortex system will induce an additional downwash over the aileron, thereby reducing its incremental lift. It is assumed that this reduction is in proportion to the reduction in lift that one would expect from a finite wing having an aspect ratio equal to the aspect ratio of that portion of the wing spanned by the aileron. If this ratio is denoted by Aa, Equation 8.96 is corrected by multiplying it by the correction to the lift curve slope for a wing.
Аа + [2(Аа + ШАа + 2)
For the case where x2 = 1.0, Aa, A, xt, and A are related by
1 -Xi +A(1 + Xt)j2
Figure 8.30b presents C(j predicted on the basis of the foregoing rela-
Figure 8.30a Simplified model of an aileron.
tionships. In preparing this figure, it was assumed that: a0~ 0.106C|/deg
7] = 0.80 (correction to t—see Equation 3.49 and Figure 3.33) jc2= 1.0
— = 0.25 c
For other aileron-wing chord ratios, this figure must be corrected for the effect on t. This correction is provided as an insert in Figure 8.30b. In order to determine C|8 for a value of cjc different from 0.25, the value of C(s read from the figure is multiplied by the factor ka. It is interesting to note that Figure 3.33 for г/ results in extreme nonlinearities in т for flap angles beyond about 15 or 20°. Yet aileron data appear linear with 5a up to angles of approximately 30°. This is further evidence that, because of induced effects, the section angles of attack are reduced over the ailerons. Indeed, according to Reference 8.8, as the aspect ratio of the portion of the wing covered by the aileron decreases, the range of Sa over which С/ is linear with Sa increases.
If x2 is less than unity, superposition can be used to calculate C, s. For example, suppose Xi = 0.5 and x2 = 0.9. Then C(s can be calculated from
Again, the subscript a is dropped on the S in writing C, s, since Sa is the primary control for producing a rolling moment.