Elementary aeroelasticity

Aircraft structures, being extremely flexible, are prone to distortion under load. When these loads are caused by aerodynamic forces, which themselves depend on the geometry of the structure and the orientation of the various structural components to the surrounding airflow, then structural distortion results in changes in aerodynamic load, leading to further distortion and so on. The interaction of aerodynamic and elastic forces is known as aeroelasticity.

Two distinct types of aeroelastic problem occur. One involves the interaction of aerodynamic and elastic forces of the type described above. Such interactions may exhibit divergent tendencies in a too flexible structure, leading to failure, or, in an adequately stiff structure, converge until a condition of stable equilibrium is reached. In this type of problem static or steady state systems of aerodynamic and elastic forces produce such aeroelastic phenomena as divergence and control reversal. The second class of problem involves the inertia of the structure as well as aerodynamic and elastic forces. Dynamic loading systems, of which gusts are of primary importance, induce oscillations of structural components. If the natural or resonant frequency of the component is in the region of the frequency of the applied loads then the amplitude of the oscillations may diverge, causing failure. Also, as we observed in Chapter 8, the presence of fluctuating loads is a fatigue hazard. For obvious reasons we refer to these problems as dynamic. Included in this group are flutter, buffeting and dynamic response.

The various aeroelastic problems may be conveniently summarized in the form of a 'tree' as follows


Static stability... Static

Dynamic ... Dynamic stability

Load distribution

Divergence Control reversal

Flutter Buffeting Dynamic response

In this chapter we shall concentrate on the purely structural aspects of aeroelasticity; its effect on aircraft static and dynamic stability is treated in books devoted primarily to aircraft stability and control12.

Redistribution of aerodynamic loads and divergence are closely related aeroelastic phenomena; we shall therefore consider them simultaneously. It is essential in the design of structural components that the aerodynamic load distribution on the component is known. Wing distortion, for example, may produce significant changes in lift distribution from that calculated on the assumption of a rigid wing, especially in instances of high wing loadings such as those experienced in manoeuvres and gusts. To estimate actual lift distributions the aerodynamicist requires to know the incidence of the wing at all stations along its span. Obviously this is affected by any twisting of the wing which may be present.

Let us consider the case of a simple straight wing with the centre of twist (or flexural centre, see Chapters 9 and 10) behind the aerodynamic centre (see Fig. 13.1). The moment of the lift vector about the centre of twist causes an increase in wing incidence which produces a further increase in lift, leading to another increase in incidence and so on. At speeds below a critical value, called the divergence speed, the increments in lift converge to a condition of stable equilibrium in which the torsional moment of the aerodynamic forces about the centre of twist is balanced by the torsional rigidity of the wing. The calculation of lift distribution then proceeds from a knowledge of the distribution of twist along the wing. For a straight wing the redistribution of lift usually causes an outward spanwise movement of the centre of pressure, resulting in greater bending moments at the wing root. In the case of a swept wing a reduction in streamwise incidence of the outboard sections due to bending deflections causes a movement of the centre of pressure towards the wing root.

All aerodynamic surfaces of the aircraft suffer similar load redistribution due to distortion.

13.1.1 Wing torsional divergence (two-dimensional case)

The most common divergence problem is the torsional divergence of a wing. It is useful, initially, to consider the case of a wing of area S without ailerons and in a

13.1 Load distribution and divergence


Wing twist

,Centre of twist

,Centre of twist

Aerodynamic centre

Fig. 13.1 Increase of wing incidence due to wing twist.

where T is the applied torque at any spanwise section z and AL and AM0 are the lift and pitching moment on the elemental strip acting at its aerodynamic centre. As 6z approaches zero, Eq. (13.4) becomes

where dc\ /da is the local two-dimensional lift curve slope and

AM0 = \PV2<r8zcmfi in which cm 0 is the local pitching moment coefficient about the aerodynamic centre. Also from torsion theory (see Chapter 3)T — GJ dd/dz. Substituting for L, M0 and T in Eq. (13.5) gives d29 {pV2ec\dcylda)e _ ~{pV2ec2(dcl/da)a {pV2c2cmfi dz2 GJ GJ GJ [ }

Equation (13.6) is a second-order differential equation in 9 having a solution of the standard form



and A and £ are unknown constants that are obtained from the boundary conditions; namely, 9 = 0 when z = 0 at the wing root and dd/dz = 0 at z = s since the torque is zero at the wing tip. From the first of these

+ a and from the second

Hence or rearranging




The smallest value corresponding to the divergence speed Vd occurs when n = 0, thus

Mathematical solutions of the type given in Eq. (13.10) rarely apply with any accuracy to actual wing or tail surfaces. However, they do give an indication of the order of the divergence speed, Vd. In fact, when the two-dimensional lift-curve slope, dci/da, is used they lead to conservative estimates of Vd. It has been shown that when dc\/da is replaced by the three-dimensional lift-curve slope of the finite wing, values of Vâ become very close to those determined from more sophisticated aerodynamic and aeroelastic theory.

The lift distribution on a straight wing, accounting for the elastic twist, is found by introducing a relationship between incidence and lift distribution from aerodynamic theory. In the case of simple strip theory the local wing lift coefficient, q, is given by in which the distribution of elastic twist 9 is known from Eq. (13.9).

13.1.3 Swept wing divergence

In the calculation of divergence speeds of straight wings the flexural axis was taken to be nearly perpendicular to the aircraft's plane of symmetry. Bending of such wings has no influence on divergence, this being entirely dependent on the twisting of the wing about its flexural axis. This is no longer the case for a swept wing where the spanwise axes are inclined to the aircraft's plane of symmetry. Let us consider the swept wing of Fig. 13.4. The wing lift distribution causes the wing to bend in an upward direction. Points A and B on a line perpendicular to the reference axis will deflect by approximately the same amount, but this will be greater than the deflection of A' which means that bending reduces the streamwise incidence of the wing. The corresponding negative increment of lift opposes the elastic twist, thereby reducing the possibility of wing divergence. In fact, the divergence speed of swept wings is so high that it poses no problems for the designer. Diederich and Budiansky in 1948

from which

Fig. 13.4 Effect of wing sweep on wing divergence speed.

showed that wings with moderate or large swcepback cannot diverge. The opposite of course is true for swept-forward wings where bending deflections have a destabilizing effect and divergence speeds are extremely low. The determination of lift distributions and divergence speeds for swept-forward wings is presented in Ref. 3.

13.2 Control effectiveness and reversal

The flexibility of the major aerodynamic surfaces (wings, vertical and horizontal tails) adversely affects the effectiveness of the corresponding control surfaces (ailerons, rudder and elevators). For example, the downward deflection of an aileron causes a nose down twisting of the wing which consequently reduces the aileron incidence. Thus, the wing twist tends to reduce the increase in lift produced by the aileron deflection, and thereby the rolling moment to a value less than that for a rigid wing. The aerodynamic twisting moment on the wing due to aileron deflection increases as the square of the speed but the elastic restoring moment is constant since it depends on the torsional stiffness of the wing structure. Therefore, ailerons become markedly less effective as the speed increases until, at a particular speed, the aileron reversal speed, aileron deflection does not produce any rolling moment at all. At higher speeds reversed aileron movements are necessary in that a positive increment of wing lift requires an upward aileron deflection and vice versa.

Similar, less critical, problems arise in the loss of effectiveness and reversal of the rudder and elevator controls. They are complicated by the additional deformations of the fuselage and tailplane-fuselage attachment points, which may be as important as the deformations of the tailplane itself. We shall concentrate in this section on the problem of aileron effectiveness and reversal.

13.2.1 Aileron effectiveness and reversal (two-dimensional case)

We shall illustrate the problem by investigating, as in Section 13.1, the case of a wing-aileron combination in a two-dimensional flow. In Fig. 13.5 an aileron deflection £ produces changes AL and AM0 in the wing lift, L, and wing pitching moment M0:

The increment of wing lift is therefore a linear function of aileron deflection and becomes zero, that is aileron reversal occurs, when

0 0

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