However, unlike the previous example, one cannot make the divergence dynamic pressure infinite or negative (thereby making divergence mathematically impossible) by choice of configuration parameters because xac/c < 1. For a given wing configuration, one is left only with the possibility of increasing the sting bending stiffness to make the divergence dynamic pressure larger.

A third configuration of a wind tunnel mount is a strut system as idealized in Figs. 3.8 and 3.9. The two linearly elastic struts have the same extensional stiffness, k, and

are mounted at the leading and trailing edges of the wing. The model is mounted in such a way as to have angle of attack of ur when the springs are both undeformed. Thus, as before, the angle of attack is a — ur + 0. As illustrated in Fig. 3.9, the elastic pitch angle, 0, can be related to the extension of the two struts as

The sum of the forces in the vertical direction shows that

The sum of the moments about the trailing edge yields

Again, using Eqs. (3.3) and (3.4) for the lift and pitching moment, the simultaneous solution of the force and moment equations yields

As usual, the divergence condition is indicated by the vanishing of the denominator, so that kc qD

It is evident for this problem as specified that, because the aerodynamic center is in front of the mid-chord, the divergence condition cannot be eliminated. However, divergence can be eliminated if the leading-edge spring stiffness is increased relative to that of the trailing-edge spring. This is left as an exercise for the reader; see Problem 5.

Undeflected wing position

Undeflected wing position

Figure 3.9 Cross section of strut-supported wind tunnel model.

3.1.4 Wall-Mounted Model for Application to Aileron Reversal

Before putting aside the wind tunnel type models dealt with so far in this chapter, we here consider the problem of aileron reversal. It is known that wing torsional flexibility causes certain primary flight control devices, such as ailerons, to function in a manner that is completely at odds with their intended purpose. The primary danger posed by the loss of control effectiveness is that the pilot cannot control the aircraft in the usual way. There are additional concerns for aircraft, the missions of which depend on their being highly maneuverable. For example, when control effectiveness is lost, the pilot may not be able to count on the aircraft's ability to execute evasive maneuvers. It is this loss in control effectiveness and eventual reversal that is the focus of this section.

To this end, consider the airfoil section of a flapped two-dimensional wing, shown in Fig. 3.10. Similar to the model discussed in Section 3.1.1, the wing is pivoted and restrained by a rotational spring with spring constant k. The main differences are that (1) a trailing-edge flap is added such that the flap angle /3 can be arbitrarily set by the flight control system; and (2) we need not consider gravity to illustrate this phenomenon, so the weight is not shown in the figure. Moment equilibrium for this system about the pivot requires that

The lift and pitching moment for a two-dimensional wing can be written as before, namely,

When ¡3 ^ 0, the effective camber of the airfoil changes, inducing changes in both lift and pitching moment. For a linear theory, both a and ¡3 should be small angles, so that

Note that CMj, < 0. For convenience, we assume a symmetric airfoil (CM„ — 0).

We now substitute Eqs. (3.25) into the moment equilibrium equation, Eq. (3.24), making use of Eqs. (3.26), and determine 0 to be

We see that, because of the torsional flexibility (represented here by the rotational spring), 0 is a function of ft. Substituting Eq. (3.27) back into Eqs. (3.25a) and (3.26a), one obtains an expression for the aeroelastic lift,

It is evident from the two terms in the coefficient of ft in this expression that lift is a function of ft in two counteracting ways. Ignoring the effect of the denominator for the time being, one sees that the first term in the numerator that multiplies ft is purely aerodynamic and leads to an increase in lift with ft, because of a change in the effective camber. The second term is aeroelastic. Recalling that CMr < 0, one sees that as ft is increased, the effective change in the camber also induces a nose-down pitching moment that, because the wing is torsionally flexible, tends to decrease 0 and in turn decrease lift. At low speed, the purely aerodynamic increase in lift overpowers the aeroelastic tendency to decrease the lift, so that the lift indeed increases with ft (and the aileron works as advertised). However, as dynamic pressure increases, the aeroelastic effect becomes stronger; and there is a point at which the net rate of change of lift with respect to ft vanishes so that c n (1 I c1SCL«CMf 3L +

Thus, one finds that the dynamic pressure at which the reversal occurs is qR = ~ rr • (3.30)

Notice that since Cm, < 0, qR > 0. Obviously, a stiffer k gives a higher reversal speed, and a torsionally rigid wing will not undergo reversal. For dynamic pressures above qR (but still below the divergence dynamic pressure), a positive ft will actually decrease the lift.

Now let us consider the effect of both numerator and denominator. As before, the denominator in L (and 0 ) can vanish, resulting in divergence. The dynamic pressure at which divergence occurs can be found by setting the denominator of Eq. (3.28) equal to zero, which yields qo = (3.31)

eSCLa

Equations (3.30) and (3.31) can be used to simplify the expression for the lift in Eq. (3.28) to obtain qS[CLar + CLp{ \-f~Jft] 1 - ^

It is clear from this expression that the coefficient of ft can be positive, negative, or zero. Thus, a positive ft could increase the lift, decrease the lift, or not change the lift at all. The aileron's lift efficiency, rj, can be thought of as the aeroelastic (i.e., actual) change in lift per unit change in ft divided by the change in lift per unit change in ft that would result were the wing not flexible in torsion; that is, change in lift per unit change in ft for elastic wing change in lift per unit change in ft for rigid wing

Using the above, one can easily find that

which implies that the wing will remain divergence-free and control efficiency will not be lost as long as q < qD < qn- Obviously, if the wing were rigid, both qD and c/r become infinite and rj — 1.

Thinking unconventionally for the moment, let us allow the possibility of qR qu- This will result in aileron reversal at a low speed, of course. Although the aileron will now work the opposite of the usual way at most of the operational speeds of the aircraft, this type of design should not be ruled out on these grounds alone. Active flight-control systems can certainly compensate for this. Moreover, one can get considerably more (negative) lift for positive ¡3 in this unusual regime than positive lift for positive p in the more conventional setting. This may have important implications for development of highly maneuverable aircraft. Exactly what other potential advantages and disadvantages exist from following this strategy, particularly in this era of composite materials, smart structures, and active controls, is not presently known and is the subject of current research.

Up to now, our aeroelastic analyses have focused on rigid wings with a flexible support. These idealized configurations do give some insight into the aeroelastic stability and response, but practical analyses must take flexibility of the lifting surface into account. That being the case, in this section we address flexible wings, albeit with simplified structural representation.

Consider an unswept uniform elastic lifting surface as illustrated in Figs. 3.11 and 3.12. The lifting surface is modeled as a beam and is presumed to be built in at its root (v = 0, to represent attachment to a wind tunnel wall or a fuselage) and free at its tip (v = £). The v axis corresponds to the elastic axis, which may be defined as the line of effective shear centers, assumed here to be straight. For isotropic beams, a transverse force applied at any point along this axis will result in bending with no elastic torsional rotation about the axis. This axis is also the axis of twist in response to a pure twisting moment applied to the wing. Because the primary concern here is the determination of the airload distributions, the only elastic deformation that will influence these loads is rotation due to twist about the elastic axis.

The total applied moment (per unit span) about this axis will be denoted as M'(y), which is positive leading-edge up and given by where L' and M'ac are the distributed spanwise lift and pitching moment (i.e., the lift and pitching moment per unit length), mg is the spanwise weight distribution (i.e., the weight per unit length), and N is the "normal load factor" for the case in which the wing is level (i.e., the z axis is directed vertically upward). Thus, N can be written as

where Az is the z component of the wing's inertial acceleration, W is the total weight of the aircraft, and L is the total lift. The distributed aerodynamic loads can be written in coefficient form as

L' — qcci, M'ac — qc2cmac, where the freestream dynamic pressure, q, is 1

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