| name | aeroelasticity |
| description | Aeroelasticity — flutter, divergence, control reversal, Theodorsen function, Vg/Vω methods, Collar's triangle, flutter speed prediction, GVT, FAR 25.629, MIL-A-8870. |
| metadata | {"priority":7,"promptSignals":{"phrases":["aeroelasticity","flutter","divergence speed","control reversal","Theodorsen function","flutter speed","wing flutter"],"minScore":3}} |
Aeroelasticity — Complete Skill
Collar's Triangle
Three force types interact:
- Aerodynamic (A): lift, drag, pitching moment (unsteady)
- Elastic (E): structural restoring forces (stiffness)
- Inertial (I): mass, moment of inertia (dynamic)
Phenomena:
- A + E → Divergence (static aeroelasticity)
- A + E + I → Flutter (dynamic aeroelasticity)
- A + E → Control reversal (static)
- A + I → Dynamic stability (no elastic effects)
Divergence (Static Aeroelasticity)
Wing divergence:
Wing twists due to lift acting ahead of elastic axis → more twist → more lift → divergence
Simple 2D wing section:
GJ × dθ/dy = e × q × c × C_Lα × (α₀ + θ) [torsion equilibrium per span]
Solution:
θ(y) = tan(λy)/λ - α₀ [λ = √(qeC_Lα / GJ)]
Divergence speed:
V_div = √[(π/2)² × GJ / (ρ × q × e × C_Lα × L_ref²)] [m/s; q = dynamic pressure at divergence]
Torsional rigidity requirement:
GJ > (q_design × e × C_Lα × L²) × (2/π)² [from divergence criterion]
Flutter
Classical flutter: coupled bending-torsion oscillation with aerodynamic energy input exceeding structural damping → divergent oscillation
Flutter speed V_F: minimum speed at which flutter occurs (must be outside design envelope)
Margin requirement (FAR 25.629): V_F ≥ 1.20 × V_D (dive speed); must demonstrate at 1.15 × V_D
Theodorsen Function (Unsteady Aerodynamics)
Lift coefficient (unsteady, oscillating airfoil):
C_L = 2π[α + ḣ/V + (3/4 - a) c/V × α̇] + π c/(2V)[ḣ/V + α̇ - a c/(V) × α̈] × C(k)
Theodorsen function C(k):
C(k) = F(k) + jG(k) [complex; k = reduced frequency = ωc/(2V)]
C(k=0) = 1 (quasi-steady); C(k→∞) = 0.5
Hankel function form:
C(k) = H₁⁽²⁾(k) / [H₁⁽²⁾(k) + j H₀⁽²⁾(k)]
Approximate values:
k = 0.05: C = 0.946 - 0.095j
k = 0.10: C = 0.906 - 0.148j
k = 0.50: C = 0.734 - 0.229j
2-DOF Flutter Equations (Bending-Torsion)
Equations of motion (per unit span):
m(ḧ + b x_α α̈) + Kh h = -L(t)
I_α α̈ + m b x_α ḧ + Kα α = M(t)
h = plunge (downward positive); α = pitch (nose up positive)
b = semi-chord; x_α = static unbalance; m = mass/span; I_α = pitch inertia/span
Flutter condition (determinant method):
Assume h(t) = h₀ e^{iωt}; α(t) = α₀ e^{iωt}
Eigenvalue problem → complex determinant = 0 → flutter frequency and speed
Vg Method (V-g Diagram)
Plot imaginary part of eigenvalue g vs. airspeed V:
g > 0 → flutter; g = 0 → flutter boundary; g < 0 → stable
Steps:
- Select frequency ratio ω/ωα
- Compute aerodynamic forces as function of k = ωb/V
- Solve 2×2 eigenvalue problem at each V → g(V) and ω(V)
- Interpolate V_flutter where g = 0
Vω Method (Alternative)
Plot ω vs. V and g vs. V; flutter at g = 0 intersection
Control Reversal
Roll control reversal:
Aileron deflection → rolling moment, but also twists wing → opposing aerodynamic moment
At reversal speed: aileron effect = zero (opposing moment = actuator moment)
Reversal speed:
V_rev = √[2 GJ × a_aileron / (ρ × c² × e × C_Lα × C_mδ × S_aileron)]
a_aileron = aileron effectiveness [per rad]
C_mδ = pitching moment per aileron deflection
Ground Vibration Test (GVT)
Purpose: measure structural natural frequencies, mode shapes, and damping before flutter analysis
Methods: single-point excitation + accelerometer arrays; multi-point random or swept sine
Key modes to measure:
- Wing bending (1st, 2nd): typically 2–10 Hz
- Wing torsion: typically 10–25 Hz
- Fuselage bending: 5–15 Hz
- Symmetric/antisymmetric combinations
Update aeroelastic model:
Adjust FEM stiffness/mass to match GVT frequencies within ±3% before flutter analysis
Flutter Testing (Flight)
Flight flutter test (FAR 25.629/MIL-A-8870):
Instrumented aircraft; excite structurally (pilot input, vane, pyrotechnic pulse)
Measure structural response; fit damping vs. speed
Subcritical flutter prediction:
Plot g vs. V; extrapolate to g = 0 → predict V_F
Require g trend shows ≥ 10% stability margin before extrapolation
Airspeed margin: V_F_predicted ≥ 1.20 × V_D (FAR 25.629)
Active Flutter Suppression
Feedback control: accelerometer or rate gyro → control surface actuator
Control law: lead-lag compensator; increases effective damping at critical flutter mode
Demonstrated on: B-2, F-16XL; wing loads reduction + flutter boundary extension
Standards
| Standard | Scope |
|---|
| FAR/CS 25.629 | Civil aircraft aeroelastic requirements |
| MIL-A-8870 | Military aircraft flutter requirements |
| MIL-A-8861 | Aeroelastic loads |
| ESDU 70004 | Flutter prediction |
Output
Provide: divergence speed V_div [m/s] and required GJ [N·m²], flutter speed V_F [m/s] and margin vs. V_D (≥ 1.20 required), critical flutter mode (bending/torsion/coupling), flutter frequency ω_F [Hz], reduced frequency k at flutter, control reversal speed V_rev [m/s], GVT target frequencies for wing bending and torsion [Hz], and applicable standard (FAR 25.629, MIL-A-8870).
