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Nonlinear dynamics of slender inverted flags in axial flow
Mohammad Tavallaeinejad, Michael P. Paidoussis, Mathias Legrand

Last modified: 2018-06-05


A nonlinear fluidelastic continuum model is proposed for a slender cantilevered plate subjected to axial flow, directed from the free end to the clamped one. A nonlinear equation of motion is derived for the plate via Hamilton’s principle in terms of the of the rotation angle of the mid-plane with respect to the axial coordinate while applying the inextensibility condition. A nonlinear unsteady slender wing theory is employed to formulate the fluid-related forces. The Galerkin scheme is utilized to discretize the system of equations, while all trigonometric terms are kept intact, resulting in reliable predictions even for very large-amplitude rotations. The numerical analysis is carried out in the time-domain using Gear's backward differentiation formula (BDF) yielding the time histories of amplitude of oscillation, together with a full Newton method yielding the static equilibria. A bifurcation diagram of the system is constructed for varying flow velocity to study the stability and post-critical behaviour of the system. It is shown that, the inverted flag is stable at its original static equilibrium position; by increasing the flow velocity, the flag initially exhibits a small-amplitude flapping around the trivial equilibrium configuration. By further increasing the flow velocity, the system displays deformed-flapping motion, (i.e. frapping motion around the deflected configuration), which changes to a fully-deflected static mode at even higher flow velocities. These predictions are in excellent agreement with existing experimental data in the literature.


inverted flag; bistability; nonlinear dynamics; fluid–structure interactions

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