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### Transcript of Aeroelastic Stability: Divergence

Aerodynamics Structural Analysis
mx(x)
d
dx
GJ(x)
d
a.c.s.c.
e
Mxac = 1
(0) = 0 GJ d
dx |x=L = 0
Aeroelasticity: Fluid-Structure Interaction (coupling)
– where:
d
dx
GJ(x)
d
dx
GJ(x)
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)
2 V 2c(cCm0 + eCl0 + eCl↵↵0)
d
dx
GJ(x)
(x) 0
Stability Solution
– Differential eigenvalue problem
d2(x)
V 2ceCl↵
(0) = 0 0(L) = 0
• Nontrivial solution only if matrix is singular
• Lowest divergence speed:
=)
=)
2
• Cantilevered • Solution:
(0) = 0 GJ d
dx |x=L = 0
2GJ (2L x)x
eCl↵
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
s
-20
-10
10
20
30
tip(deg)