The origin of hysteresis in the flag instability

Abstract

The flapping flag instability occurs when a flexible cantilevered plate is immersed in a uniform airflow. To this day, the nonlinear aspects of this aeroelastic instability are largely unknown. In particular, experiments in the literature all report a large hysteresis loop, while the bifurcation in numerical simulations is either supercritical or subcritical with a small hysteresis loop. In this paper, this discrepancy is addressed. First weakly nonlinear stability analyses are conducted in the slender-body and two-dimensional limits, and second new experiments are performed with flat and curved plates. The discrepancy is attributed to inevitable planeity defects of the plates in the experiments.

The effect of non-uniform damping on flutter in axial flow and energy-harvesting strategies

“…The critical flow speed at which flutter ensues is verified from linear stability analysis and is confirmed with Païdoussis et al (2002). Consistent with flutter in plates (Eloy et al 2012), we note that this instability is a supercritical Hopf bifurcation with flow speed. The jumps in the bifurcation curve correspond to the mode switching reported in Semler et al (2002); this may also be discerned from the snapshots of the beam at three different flow speeds.…”

“…A Kelvin-Voigt damping model is considered here, generalizing previous contributions (Païdoussis 2004;Tang et al 2009;Doaré 2010;Eloy et al 2012) to non-uniform damping distributions.…”

“…This reactive force does not include any flow separation associated with the transverse motion of each cross section, and, as emphasized in Candelier et al (2011), for freely flapping bodies, an additional contribution for the fluid force must be included to account for such dissipative effects. Here, an empirical 'resistive' force model is therefore added following Taylor (1952) and Eloy et al (2012):…”

Passive elastic mechanism to mimic fish-muscle action in anguilliform swimming

“…This so-called reactive force results from the reaction of the fluid accelerated by the body movements and can be expressed as [32,33] …”

Theory and experiment for flutter of a rectangular plate with a fixed leading edge in three-dimensional axial flow

“…For the linear analysis presented here the only data that the model should be compared to are the unfilled squares because the gap in the flutter velocity down to the filled in squares represents a hysteretic effect which is not captured by the current linear model. A recent publication suggests that the hysteresis arises due to spanwise deformations in the structure (Eloy et al, 2012;Zhao et al, 2011) before the onset of flutter which become less important once the structure begins to flutter. This may explain why the current theoretical predictions match the lower flutter velocities as they are observed after these deformations have been eliminated by a violent flutter motion.…”