Bifurcation analysis of cantilever beams in channel flow

Abstract

The flutter of cantilevered beams in channel flow is a benchmark example of flow-induced vibrations and its fundamental behaviour is found in numerous practical applications. Experiments have shown that such systems present a wide variety of complex nonlinear behaviour. However, the plethora of previous studies is mostly concerned with linear stability analysis. In this work, we provide an initial impulse for a comprehensive nonlinear study of these systems through bifurcation analysis. We consider a one-dimensional problem, where a cantilevered beam is treated in a modal framework and the surrounding flow is modelled by bulk-flow equations. The system is discretized in space and time via Galerkin procedures (modal, Tau and harmonic balance) and the continuation of periodic solutions is achieved using the asymptotic numerical method. Additionally, a numerical method for an “augmented” linear stability analysis is proposed, allowing the continuation of Hopf bifurcation branches, including their sub- or super-critical nature. The nonlinear dynamics are explored with respect to various dimensionless parameters. Results illustrate a number of behavioural trends: sub-critical bifurcations and hysteresis loops, internal resonances, grazing boundaries (separation between limit cycles with and without intermittent beam-wall impacts) as well as torus bifurcations and quasi-periodic oscillations.