Divergence occurs when the smallest eigenvalue (\lambda_\min) of (\mathbfK^-1 \mathbfA 0) satisfies (q \infty, \textdiv = 1 / \lambda_\min). Physically, aerodynamic moments overcome structural stiffness. Assume harmonic motion (\mathbfu = \hat\mathbfu e^i\omega t) and use frequency-domain aerodynamics (\mathbfQ(i\omega)):
typically uses a loose staggering with sub-iterations: theoretical and computational aeroelasticity pdf
[ \left[ -\omega^2 \mathbfM + i\omega \mathbfC + \mathbfK - q_\infty \mathbfQ(i\omega) \right] \hat\mathbfu = 0 ] flutter (dynamic instability)
1. Introduction Aeroelasticity studies the mutual interaction among aerodynamic, elastic, and inertial forces. Its theoretical foundation enables prediction of critical phenomena: divergence (static instability), flutter (dynamic instability), and buffeting (forced response). Computational aeroelasticity extends these theories into numerical solvers that couple structural dynamics with aerodynamic models—ranging from potential flow to large-eddy simulation (LES). 2. Theoretical Framework: The Aeroelastic Governing Equation For a linear structure discretized via finite elements, the semi-discrete equations of motion are: \textdiv = 1 / \lambda_\min). Physically
[ \mathbfM\ddot\mathbfu + \mathbfC\dot\mathbfu + \mathbfK\mathbfu = q_\infty \left( \mathbfA_0 \mathbfu + \mathbfA_1 \dot\mathbfu + \int_0^t \mathbfG(t-\tau)\dot\mathbfu(\tau) d\tau \right) ]
[ \mathbfK \mathbfu = q_\infty \mathbfA_0 \mathbfu ]
V_range = np.linspace(50, 300, 50) # velocity (m/s) b = 0.5 # reference semi-chord