Question 1.8.2: The vibration of an aircraft wing can be crudely modeled as ......
The vibration of an aircraft wing can be crudely modeled as
m \ddot{x} + c\dot{x} + kx = \gamma \dot{x}where m, c, and k are the mass, damping, and stiffness values of the wing, respectively, modeled as a single-degree-of-freedom system, and where the term \gamma \dot{x} is an approximate model of the aerodynamic forces on the wing (\gamma > 0 for high speed).
Rearranging the equation of motion yields
m\ddot{x} + (c + \gamma) \dot{x} + kx = 0
If \gamma and c are such that c – \gamma > 0, the system is asymptotically stable. However, if \gamma is such that c – \gamma < 0, then ζ = (c – \gamma)/2mw_{n} < 0 and the solutions are of the form
x(t) = Ae^{- \zeta w_{n} t} \sin(w_{d}t + Φ)
where the exponent (–ζw_{n}t) > 0 for all t > 0 because of the negative damping term. Such solutions increase exponentially with time, as indicated in Figure 1.39. This is an example of flutter instability and self-excited oscillation.
