aeroelasticity

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:

1. Select frequency ratio ω/ωα
2. Compute aerodynamic forces as function of k = ωb/V
3. Solve 2×2 eigenvalue problem at each V → g(V) and ω(V)
4. 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).