Saving energy and enhancing performance are secular preoccupations shared by both nature and human beings. In animal locomotion, flapping flyers or swimmers rely on the flexibility of their wings or body to passively increase their efficiency using an appropriate cycle of storing and releasing elastic energy. Despite the convergence of many observations pointing out this feature, the underlying mechanisms explaining how the elastic nature of the wings is related to propulsive efficiency remain unclear. Here we use an experiment with a self-propelled simplified insect model allowing to show how wing compliance governs the performance of flapping flyers. Reducing the description of the flapping wing to a forced oscillator model, we pinpoint different nonlinear effects that can account for the observed behavior-in particular a set of cubic nonlinearities coming from the clamped-free beam equation used to model the wing and a quadratic damping term representing the fluid drag associated to the fast flapping motion. In contrast to what has been repeatedly suggested in the literature, we show that flapping flyers optimize their performance not by especially looking for resonance to achieve larger flapping amplitudes with less effort, but by tuning the temporal evolution of the wing shape (i.e., the phase dynamics in the oscillator model) to optimize the aerodynamics. F lying animals have long since inspired admiration and fueled the imagination of scientists and engineers. Alongside biologists studying form and function of flapping flyers in nature (1, 2), the last decade has seen an impressive quantity of studies driven by engineering groups using new techniques to develop and study artificial biomimetic flapping flyers (3, 4). The widespread availability of high-speed video and in particular the merging of experimental methods borrowed from fluid mechanics into the toolbox of the experimental biologist have permitted to elucidate various key mechanisms involved in the complex dynamics of flapping flight (see, for example, refs. 5-7).A recent field of investigation concerns the efficiency of flapping flyers, the major interrogation being about how natural systems optimize energy saving together with performance enhancement. In particular, the passive role of wing flexibility to increase flight efficiency through the bending of flapping wings has attracted a lot of attention. It is commonly agreed that this efficiency enhancement comes from the particular shape of the bent wing, which leads to a more favorable repartition of the aerodynamic forces (see refs. 8 and 9 for an extensive review). For flying animals in air, such as insects, it has been proposed (10-12) that wing inertia should play a major role in competing with the elastic restoring force, compared to the fluid loading. The mechanism governing the propulsive performance of the flapping flyer can therefore be seen at leading order as a two-step process, where the instantaneous shape of the wings is determined by a structural mechanics problem that then sets...

We study experimentally the vortex streets produced by a flapping foil in a hydrodynamic tunnel, using 2D Particle Image Velocimetry (PIV). An analysis in terms of a flapping frequency-amplitude phase space allows to identify: 1) the transition from the well-known B\'enard-von K\'arm\'an (BvK) wake to the reverse BvK vortex street that characterizes propulsive wakes, and 2) the symmetry breaking of this reverse BvK pattern giving rise to an asymmetric wake. We also show that the transition from a BvK wake to a reverse BvK wake precedes the actual drag-thrust transition and we discuss the significance of the present results in the analysis of flapping systems in nature.Comment: Revised version includes minor changes in the discussion of the drag coefficient calculation. The 4 figures are unchange