AOE 3124
Aeroelasticity and Wing Divergence
Aerospace Structures
The following paragraphs are extracted from the text Aeroelasticity, by R.L. Bisplinghoff, H. Ashley, and R.L. Halfman, Dover Publications, Inc., Mineola, New York, 1996, pp. 1-3 (originally published by Addison-Wesley, 1955).
Aeroelasticity is defined as a science which studies the mutual interaction between aerodynamic forces and elastic forces, and the influence of this interaction on airplane design. Aeroelastic problems would not exist if the airplane structure were perfectly rigid. Modern airplane structures are very flexible, and this flexibility is
fundamentally responsible for the various types of aeroelastic phenomena. Structural flexibility itself may not be
objectionable; however, aeroelastic phenomena arise when structural deformations induce additional aerodynamic forces. Such interactions may become smaller and smaller until a condition of stable equilibrium is
reached, or they may tend to diverge and destroy the structure.
The term aeroelasticity, however, is not completely descriptive, since many important aeroelastic phenomena
involve inertial forces as well as aerodynamic and elastic forces. We shall apply a definition in which the term
aeroelasticity includes phenomena involving interactions among inertial, aerodynamic, and elastic forces, and
other phenomena involving interactions between aerodynamic and elastic forces. The former will be referred to
as dynamic and the latter as static aeroelastic phenomena.
Divergence is a static instability of a lifting surface of an aircraft in flight, at a speed called the divergence speed,
where the elasticity of lifting surface plays an essential role in the instability.
Divergence analysis of a rigid wing segment
A model to illustrate the phenomenon of wing divergence consists of a uniform, rigid wing segment hinged to a
fixed support in a wind tunnel as is shown in Figure 1. The hinge line is located at the elastic axis (E.A.) of the wing.
The elastic axis coincides with the locus of shear centers of the wing sections. Recall that the shear center of the
cross section of a bar (wing) is a reference point in the cross section where the lateral deflections due to bending are
de-coupled from the twist due to torsion; i.e., a shear force acting at the elastic axis results in bending deflections
and no twist, and a torque acting at the elastic axis causes twist but no lateral deflection of the elastic axis due to
bending. The rigid wing segment is restrained against rotation, or twist, about the E.A. by a linear elastic rotational
spring of stiffness K . This rotational spring is analogous the torsional stiffness per unit span, or ( GJ ) ⁄ L , of a wing.
L
A.C.
M AC
C.G.
V
E.A.
α
FIGURE 1.
W
K
d
e
We assume two-dimensional, incompressible aerodynamics is applicable. Let V denote the airspeed, α the angle of attack relative to the zero lift angle, L the lift, M AC the pitching moment, and let W denote the weight of the
wing segment acting at the center of gravity (C.G.). The lift and pitching moment act at the aerodynamic center
(A.C.), which is the point about which the pitching moment is independent of the angle of attack. Usually the A.C. is
close to the quarter chord. We neglect the drag force D relative to the lift since D « L . The angle of attack is written as
Page 1 of 6
AOE 3124
Aeroelasticity and Wing Divergence
Aerospace Structures
α = α0 + θ
( 1)
where α 0 is the initial wing incidence, or the angle of attack if there are no aerodynamic and gravity loads, and θ is
the rotation angle due to elastic deformations of the spring. Assume small angles such that sin α ≈ α , cos α ≈ 1 , and
tan α ≈ α . The lift is given by
L = qSC L
( 2)
where q is the dynamic pressure, S is the wing planform area, and C L is the lift coefficient. Let ρ denote the density of air, c the chord length, and let s denote the wing span. The dynamic pressure, planform area, and lift coefficient are given by
1
q = --- ρV 2
2
S = cs
∂C
C L = ⎛ ---------L⎞ α
⎝ ∂α ⎠
( 3)
In the above equation ( ∂C L ) ⁄ ( ∂α ) is the lift curve slope, which is assumed constant between stall points. The
pitching moment is given by
M AC = qScC MAC
( 4)
where C MAC is the pitching moment coefficient about the A.C., and is independent of α .
Moment equilibrium about the E.A. gives
eL + M MAC – Wd – Kθ = 0
( 5)
where e is the distance from the E. A. to the A. C., assuming the E.A. is behind the A.C. Substituting for the elastic
twist, lift, and pitching moment from equations (1) to (4), the moment equation becomes
∂C
eqS ⎛ ---------L⎞ α + qScC MAC – Wd – K ( α – α 0 ) = 0
⎝ ∂α ⎠
( 6)
Rearrange this last equation to
∂C
qS ⎛ ---------L⎞ e – K α = Wd – qScC MAC – Kα 0
⎝ ∂α ⎠
Now divide by – K to get
∂C
qS ⎛ ---------L⎞ e
⎝ ∂α ⎠
Wd qScC MAC
1 – ------------------------- α = α 0 – -------- + ---------------------K
K
K
Let
K
q D = --------------------∂C
S ⎛ ---------L⎞ e
⎝ ∂α ⎠
Hence, equation (7) for the equilibrium value of the angle of attack reduces to
Page 2 of 6
( 7)
AOE 3124
Aeroelasticity and Wing Divergence
Aerospace Structures
q c C MAC
α 0 – ( Wd ) ⁄ K + ⎛ ------⎞ ⎛ --⎞ -------------⎝ q D⎠ ⎝ e⎠ ∂C
⎛ ---------L⎞
⎝ ∂α ⎠
α = ---------------------------------------------------------------------------------1 – q ⁄ qD
( 8)
A plot of q ⁄ q D versus the angle of attack obtained from equation (8) is shown in Figure 2.
q
-----qD
1
FIGURE 2.
0
α
α 0 – ( Wd ) ⁄ K
From equation (8) we see that α → ∞ as q → q D for 0 ≤ q < q D . That is, the angle of attack grows without bound
as q → q D . Of course, this excessive twist is a theoretical result. In reality the wing with stall or twist off due to a
strength failure. Hence, the divergence dynamic pressure is defined as
K
q D = --------------------∂C
S ⎛ ---------L⎞ e
⎝ ∂α ⎠
( 9)
and the divergence speed is
VD =
2q D
--------ρ
( 10)
Divergence corresponds to static instability. At V = V D we get excessive rotation in twist of the wing.
Straight, uniform, unswept, high aspect ratio, cantilevered wing in steady incompressible flow
c
V
line of A.C.s
1
q = --- ρV 2
2
t z dz
T
T + dT
dz
α
L
differential element
e
dz
z E.A.
FIGURE 3.
Page 3 of 6
AOE 3124
Aeroelasticity and Wing Divergence
Aerospace Structures
Let α ( z ) denote the total wing incidence, and let α 0 denote the fixed incidence at the wing root. The fixed incidence could be a function of z for a variable in built-in twist, but we will consider it constant along the span. So
α = α0 + θz ( z )
( 11)
where θ z ( z ) is the twist angle of the wing due to elastic deformation. Neglect airfoil weight, since we saw for the
rigid wing segment that this factor played no role in the divergence dynamic pressure. From St. Venant torsion theory we have the differential equation of torque as
dT
------ + t z = 0
dz
0<z<L
( 12)
where t z denotes the external torque per unit span. (Also, refer to eq. (7.22) on page 1461.) In this case the external
torque per unit span is due to the aerodynamic loads acting on the wing. St. Venant’s torsion theory relates the torque
to the unit twist as
T = GJ
dθ z
dz
( 13)
where GJ is the torsional stiffness of the wing box (See eq. (7.24) page 1471). Recall from eq. (7.51) on page 1581
that the torsion constant for a single cell, homogeneous cross section is given by
4A 2
J = --------cds
----t
°∫
where A c is the area enclosed by the cross-sectional contour, s is the arc length along the contour, and t is the
thickness of the contour. Substitute equation (13) into (12) and use the fact that the wing is uniform along the span to
get
2
dθ
GJ 2 z + t z = 0
dz
θz = θz ( z )
0<z<L
( 14)
Aerodynamics. Use “strip theory”, which assumes aerodynamic lift and moment at station z depends only on the
angle of attack, or incidence, at z , and is independent of the angle of attack at any other spanwise locations. Physically, this is reasonable for high aspect ratio wings.
⎧
⎨
⎩
dL = q ( cdz ) c l
c
dS
⎧
⎨
⎩
dM AC = qc ( cdz ) c mAC
FIGURE 4.
dS
dz
e
A.C.
E.A.
The differential lift and differential pitching moment acting at the A.C. on an typical element of the wing are shown
in the Figure 4, where c l is the local lift coefficient and c mAC is the local moment coefficient about the A.C. Hence,
the external torque acting on the differential element about the elastic axis is
1.
Thin-Walled Structures, E.R. Johnson, Fall 2003, AOE 3024 text
Page 4 of 6
AOE 3124
Aeroelasticity and Wing Divergence
Aerospace Structures
t z dz = edL + dM AC = [ eqcc l + qc 2 c mAC ]dz
( 15)
t z ( z ) = qcec l + qc 2 c mAC
( 16)
∂c
c l = ⎛ l⎞ α = a 0 α
⎝ ∂ α⎠
( 17)
or
According to strip theory
where a 0 is the lift curve slope. Substituting equation (11) into (17), we get
cl = a0 [ α0 + θz ( z ) ]
( 18)
t z ( z ) = qcea 0 [ α 0 + θ z ( z ) ] + qc 2 c mAC
( 19)
Hence,
Now substitute equation (19) into (14) and rearrange the terms to get
2
dθ
GJ 2 z + ( qcea 0 )θ z = – qcea 0 α 0 – qc 2 c mAC
dz
( 20)
Equation (20) is the governing, second order, ordinary differential equation for θ z ( z ) with 0 < z < L . The boundary
conditions at z = 0 and z = L are to specify either θ z or T , but not both. For a cantilever wing, which is clamped
at the root and free at the tip, the boundary conditions are
θz ( 0 ) = 0
T ( L ) = GJ
dθ z
dx
= 0
( 21)
z=L
The general solution of the o.d.e. (20) is the sum of a particular solution and a homogenous solution.
θz ( z ) = θp ( z ) + θh ( z )
By the method of undetermined coefficients the particular solution is
cc mAC
θ p = – α 0 – -------------.
ea 0
2
The solution of the homogenous equation GJ
d θh
+ ( qcea 0 )θ h = 0 is
dz2
θ h = A 1 sin ( λz ) + A 2 cos ( λz ) .
where
λ 2 = ( qcea 0 ) ⁄ ( GJ )
( 22)
Hence, the general solution for the wing twist is
cc mAC
θ z ( z ) = A 1 sin ( λz ) + A 2 cos ( λz ) – α 0 – -------------ea 0
Page 5 of 6
( 23)
AOE 3124
Aeroelasticity and Wing Divergence
Aerospace Structures
Substitute the general solution (23) into the boundary conditions (21) to determine constants A 1 and A 2 :
cc mAC
A 2 – α 0 – -------------= 0
ea 0
A 1 λ cos ( λL ) – A 2 λ sin ( λL ) = 0
Solving the latter equations for the constants, we get
cc mAC⎞
A 1 = ⎛ α 0 + -------------tan ( λL )
⎝
ea 0 ⎠
cc mAC
A 2 = α 0 + -------------ea 0
( 24)
Substitute the above results for the constants in equation (23) to get the angle of twist for the cantilevered wing as
cc mAC⎞
θ z ( z ) = ⎛ α 0 + -------------[ tan ( λL ) sin ( λz ) + cos ( λz ) – 1 ]
⎝
ea 0 ⎠
( 25)
Hence from equations (11) and (25), the total wing incidence is
cc mAC⎞ sin ( λL ) sin ( λz ) + cos ( λL ) cos ( λz )
α ( z ) = α 0 + ⎛ α 0 + --------------------------------------------------------------------------------------------------- – 1
⎝
ea 0 ⎠
cos ( λL )
( 26)
Using the trigonometric identity for the cosine of the sum of two angles, this last result can be written as
cc mAC⎞ cos [ λ ( L – z ) ]
α ( z ) = α 0 + ⎛ α 0 + ----------------------------------------------- – 1
⎝
cos ( λL )
ea 0 ⎠
( 27)
From equation (27) we see that α → ∞ if cos ( λL ) = 0 . Vanishing of the cosine function occurs when
π π
λL = ---, 3 ---, … The lowest root gives the critical divergence dynamic pressure as
2 2
⎛λ L = π
--- =
⎝ D
2
q D cea 0 ⎞
π 2 GJ
------------------ L → q D = ⎛ ------⎞ ----------⎝ 2L⎠ cea 0
GJ ⎠
( 28)
The value of q D in equation (28) is the wing’s torsional divergence dynamic pressure.
The analogy between the divergence dynamic pressure for the rigid wing segment model, equation (9), and the
elastic wing divergence dynamic pressure, equation (28), is tabulated below.
Rigid
Elastic
K
( GJ ) ⁄ L
S
Lc
e
e
( ∂C L ) ⁄ ( ∂α )
a0
Page 6 of 6