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A generalized theory of viscous and inviscid flutter. (English) Zbl 1195.76450
Summary: We present a unified theory of flutter in inviscid and viscous flows interacting with flexible structures based on the phenomenon of $$1:1$$ resonance. We show this by treating four extreme cases corresponding to viscous and inviscid flows in confined and unconfined flows. To see the common mechanism clearly, we consider the limit when the frequencies of the first few elastic modes are closely clustered and small relative to the convective fluid time scale. This separation of time scales slaves the hydrodynamic force to the instantaneous elastic displacement and allows us to calculate explicitly the dependence of the critical flow speed for flutter on the various problem parameters. We show that the origin of the instability lies in the coincidence of the real frequencies of the first two modes at a critical flow speed beyond which the frequencies become complex, thus making the system unstable to oscillations. This critical flow speed depends on the difference between the frequencies of the first few modes and the nature of the hydrodynamic coupling between them. Our generalized framework applies to a range of elastohydrodynamic systems and further extends the Benjamin-Landahl classification of fluid-elastic instabilities.

##### MSC:
 76Z10 Biopropulsion in water and in air 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
##### Keywords:
fluid-structure interaction; flutter; $$1:1$$ resonance
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