Redistribution of aerodynamic loads and divergence are closely related aeroelastic phe- nomena; 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 distribu- tion 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 behind the aerodynamic centre (see Fig. 28.1). 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.
28.2.1 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
Fig. 28.1Increase of wing incidence due to wing twist.
28.2 Load distribution and divergence 747 two-dimensional flow, as shown in Fig. 28.2. The torsional stiffness of the wing, which we shall represent by a spring of stiffness, K, resists the moment of the lift vector, L, and the wing pitching moment, M0, acting at the aerodynamic centre of the wing section. For moment equilibrium of the wing section about the aerodynamic centre we have
M0+Lec=Kθ (28.1)
where ec is the distance of the aerodynamic centre forward of the flexural centre expressed in terms of the wing chord, c, andθis the elastic twist of the wing. From aerodynamic theory
M0 = 1
2ρV2ScCM,0 L= 1
2ρV2SCL Substituting in Eq. (28.1) yields
1
2ρV2S(cCM,0+ecCL)=Kθ or, since
CL =CL,0+∂CL
∂α (α+θ)
in whichαis the initial wing incidence or, in other words, the incidence corresponding to given flight conditions assuming that the wing is rigid and CL,0 is the wing lift coefficient at zero incidence, then
1 2ρV2S
cCM,0+ecL,0+ec∂CL
∂α (α+θ)
=Kθ where∂CL/∂αis the wing lift curve slope. Rearranging gives
θ
K−1
2ρV2Sec∂CL
∂α
= 1 2ρV2Sc
CM,0+eCL,0+e∂CL
∂α α
Fig. 28.2Determination of wing divergence speed (two-dimensional case).
or
θ=
1
2ρV2Sc[CM,0+eCL,0+e(∂CL/∂α)α]
K−12ρV2Sec(∂CL/∂α) (28.2) Equation (28.2) shows that divergence occurs (i.e.θbecomes infinite) when
K = 1
2ρV2Sec∂CL
∂α The divergence speed Vdis then
Vd=
2K
ρSec(∂CL/∂α) (28.3)
We see from Eq. (28.3) that Vdmay be increased either by stiffening the wing (increasing K) or by reducing the distance ec between the aerodynamic and flexural centres. The former approach involves weight and cost penalties so that designers usually prefer to design a wing structure with the flexural centre as far forward as possible. If the aerodynamic centre coincides with or is aft of the flexural centre then the wing is stable at all speeds.
28.2.2 Wing torsional divergence (finite wing)
We shall consider the simple case of a straight wing having its flexural axis nearly perpendicular to the aircraft’s plane of symmetry (Fig. 28.3(a)). We shall also assume that wing cross-sections remain undistorted under the loading. Applying strip theory
Fig. 28.3Determination of wing divergence speed (three-dimensional case).
28.2 Load distribution and divergence 749 in the usual manner, i.e. we regard a small element of chord c and spanwise widthδz as acting independently of the remainder of the wing and consider its equilibrium, we have from Fig. 28.3(b), neglecting wing weight
T+ dT
dzδz
−T+Lec+M0=0 (28.4)
where T is the applied torque at any spanwise section z andL andM0are the lift and pitching moment on the elemental strip acting at its aerodynamic centre, respectively.
Asδz approaches zero, Eq. (28.4) becomes dT
dz +ecdL
dz + dM0
dz =0 (28.5)
In Eq. (28.4)
L= 1
2ρV2cδz∂c1
∂α(α+θ) where∂c1/∂αis the local two-dimensional lift curve slope and
M0 = 1
2ρV2c2δzcm,0
in which cm,0is the local pitching moment coefficient about the aerodynamic centre.
Also from torsion theory (see Chapter 3) T=GJ dθ/dz. Substituting for L, M0 and T in Eq. (28.5) gives
d2θ dz2 +
1
2ρV2ec2(∂c1/∂α)θ
GJ = −12ρV2ec2(∂c1/∂α)α
GJ −
1
2ρV2c2cm,0
GJ (28.6)
Equation (28.6) is a second-order differential equation in θ having a solution of the standard form
θ=A sinλz+B cosλz−
cm,0 e(∂c1/∂α)+α
(28.7) where
λ2=
1
2ρV2ec2(∂c1/∂α) GJ
and A and B are unknown constants that are obtained from the boundary conditions;
namely,θ=0 when z=0 at the wing root and dθ/dz=0 at z=s since the torque is zero at the wing tip. From the first of these
B=
cm,0 e(∂c1/∂α) +α
and from the second
A=
cm,0 e(∂c1/∂α) +α
tanλs
Hence
θ=
cm,0 e(∂c1/∂α)+α
( tanλs sinλz+cosλz−1) (28.8) or rearranging
θ=
cm,0
e(∂c1/∂α)+α cosλ(s−z) cosλs −1
(28.9) Therefore, at divergence when the elastic twist,θ, becomes infinite
cosλs=0 so that
λs=(2n+1)π
2 for n=0, 1, 2,. . .,∞ (28.10) The smallest value corresponding to the divergence speed Vdoccurs when n=0, thus
λs=π/2 or
λ2=π2/4s2 from which
Vd=
π2GJ
2ρec2s2(∂c1/∂α) (28.11)
Mathematical solutions of the type given in Eq. (28.10) rarely apply with any accur- acy 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,∂c1/∂α, is used they lead to conservative estimates of Vd. It has been shown that when∂c1/∂αis replaced by the three-dimensional lift-curve slope of the finite wing, values of Vdbecome 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, c1, is given by
c1= ∂c1
∂α(α+θ)
in which the distribution of elastic twistθis known from Eq. (28.9).
28.2.3 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