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Aeroelastic Stability: Divergence
Analytical ExactLater: Approximate
Structural Analysis: Aeroelastic Divergence© 2021 Mayuresh Patil. Licensed under a Creative Commons Attribution 4.0 license https://creativecommons.org/licenses/by-nc-sa/4.0/ [email protected]
Aeroelasticity
• Aeroelastic phenomena– Stability: Divergence (static) and Flutter (Dynamic)– Lift redistribution– Control reversal
AerodynamicsStructural Analysis
Undeformed AiplaneFlight Condition
Airloads DeformationStresses
Aeroelastic Coupling
Torsion (review)
• Equations of Equilibrium
• Internal Twisting Moment
• Strain and Stress (Axisymmetric)
• Strain and Stress (Thin-Walled, Single Cell)
mx(x)
d
dx
GJ(x)
d�(x)
dx
�= �mx(x)
Tx(x) = GJ(x)d�(x)
dx
⌧xs =Tx
2Act✏xs =
Ac
tRc1t ds
d�
dxJ =
4A2cR
c1t ds
⌧x✓ = Grd�
dx=
Txr
J✏x✓ =
1
2rd�
dxJ = Ip
Torsion of Wing
• Aerodynamic pitching moment:
– Twist of the wing due to aerodynamic loads
a.c.s.c.
e
mx = Mxsc = Mxac + Ly ⇥ e
Mxac =1
2⇢V 2c2Cm0
Ly =1
2⇢V 2cCl =
1
2⇢V 2c(Cl0 + Cl↵↵)
d
dx
GJ(x)
d�(x)
dx
�= �1
2⇢V 2c(cCm0 + eCl0 + eCl↵↵)
�(0) = 0 GJd�
dx|x=L = 0
Aeroelasticity:Fluid-Structure Interaction (coupling)
• Aerodynamic loads lead to structural deformation (φ)• Does structural deformation affect the aerodynamic loads?
– where:
• Aeroelastic equation for torsion of wing↵ = ↵0 + �(x)
d
dx
GJ(x)
d�(x)
dx
�= �1
2⇢V 2c(cCm0 + eCl0 + eCl↵(↵0 + �(x)))
d
dx
GJ(x)
d�(x)
dx
�+
1
2⇢V 2ceCl↵�(x) = �1
2⇢V 2c(cCm0 + eCl0 + eCl↵↵0)
Ly =1
2⇢V 2c(Cl0 + Cl↵↵)
Wing twist under aerodynamic loads• We can calculate the twist at a given flight condition
• Possibility of Instability? Yes • Large twist at certain airspeed: Divergence– Consider solutions of homogenous equation
– Trivial solution:– Is there a non-trivial solution?
d
dx
GJ(x)
d�(x)
dx
�+
1
2⇢V 2ceCl↵�(x) = �1
2⇢V 2c(cCm0 + eCl0 + eCl↵↵0)
d
dx
GJ(x)
d�(x)
dx
�+
1
2⇢V 2ceCl↵�(x) = 0
�(x) ⌘ 0
Stability Solution
• Uniform wing (constant GJ, c, e, Clα)
– Differential eigenvalue problem
• Solution:• Example: Cantilevered wing
GJd2�(x)
dx2+
1
2⇢V 2ceCl↵�(x) = 0
d2�(x)
dx2+ �2�(x) = 0 �2 =
⇢V 2ceCl↵
2GJ
�(x) = A sin�x+B cos�x
�(0) = 0 �0(L) = 0
Algebraic Eigenvalue problem• Matrix form
• Nontrivial solution only if matrix is singular
• Lowest divergence speed:
– No divergence if e < 0
=)
=)
0 1
cos�L � sin�L
�⇢AB
�= 0
cos�L = 0 �L =(2n� 1)⇡
2
�2 =(2n� 1)2⇡2
4L2Vdiv =
(2n� 1)⇡
2
s2GJ
⇢ceCl↵L2
Vdiv =⇡
2
s2GJ
⇢ceCl↵L2
qdiv =⇣⇡2
⌘2 GJ
ceCl↵L2
Uniform Wing:twist at a given flight conditions
• Cantilevered• Solution:
– What happens when– Note: if no aeroelastic coupling:
�(0) = 0 GJd�
dx|x=L = 0
�(x) =qc(cCm0 + e(Cl0 + Cl↵↵0))
2GJ(2L� x)x
q !⇣⇡2
⌘2 GJ
ceCl↵L2
�(x) =cCm0 + e(Cl0 + Cl↵↵0)
eCl↵
"�1 + cos
rqceCl↵
GJx
!+ sin
rqceCl↵
GJx
!tan
rqcL2eCl↵
GJ
!#
GJd2�(x)
dx2+ qceCl↵�(x) = �qc(cCm0 + eCl0 + eCl↵↵0)
Aeroelastic Wing Twist
• GJ = 0.26e9 N-m2, L = 25 m, ρ=1 kg/m3, c = 4 m, e = 1 m, Clα = 2 π, Cl0= 0.25, Cm0 = 0, α0 = 3 deg
• Vdiv = 285.8 m/s
50 100 150 200 250 300Vm
s
-20
-10
10
20
30
�tip(deg)
Change in Lift due to Flexibility