A nonlinear analytical formulation for the 1D modelling of a flexible beam in channel flow

Abstract

The dynamics of flexible beams in confined flows has been a subject of research for many years, as its fundamental behaviour is found in many applications, from energy harvesters to musical instruments. Most studies are concerned solely with the conditions for linear stability and do not explore the ensuing nonlinear behaviour of the system. This is particularly delicate as fluttering beams in confined flows are known to often result in dynamics with intermittent impacts between the beam and the side-walls. Here we present a nonlinear analytical resolution to a simplified 1-D model, based on a modal beam and bulk-flow equations. The model accounts for dissipation through distributed frictional and localized​ head-loss terms. The latter are imposed at the boundary conditions and aim to describe the complex effects occurring outside the domain (turbulence, vortex shedding, etc.). The present analytical resolution leads to a compact system for linear stability analysis, but also to a nonlinear formulation of the fluid–structure interaction. The inclusion of a regularized contact model allows for the computation of the full nonlinear dynamics, including intermittent impacts. Linear stability results are compared to previously published results using 2-D CFD models, and the relative merits of the model are discussed. A variety of limit cycles, (1) with and without impacts, (2) in symmetric and asymmetric configurations and (3) with impacts both at the beam tip and along its length, are shown to illustrate the diversity of dynamics encountered. Moreover, we show that, at large flow velocities, particular model configurations can lead to aperiodic dynamics, a phenomenon reported in several experimental observations. To the authors knowledge, the proposed formulation presents, for the first time, a framework for the comprehensive understanding of the nonlinear dynamics associated with flexible beams in confined axial flow.