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A New Vibrational Control System in Nature: Flapping Flight

Vibrational control is an open loop stabilization technique via the application of highamplitude, high-frequency oscillatory inputs. The averaging theory has been the standard technique for designing vibrational control systems. However, it stipulates too high oscillation frequency that may not be practically feasible. Therefore, although vibrational control is very robust and elegant (stabilization without feedback), it is rarely used in practical applications. The only well-known example is the Kapitza pendulum; an inverted pendulum shose pivot is subject to vertical oscillation. the unstable equilibrium of the inverted pendulum gains asymptotic stability due to the high-frequency oscillation of the pivot. In this paper, we provide a new vibrational control system from Nature; flapping flight dynamics. Flapping flight is a rich dynamical system as a representative model will typically be nonlinear, time-varying, multi-body, multi-time-scale dynamical system. Over the last two decades, using direct averaging, there has been consensus in the flapping flight dynamics community that insects are unstable at the hovering equilibrium due to the lack of pitch stiffness. In this work, we perform higher-order averaging of the time-periodic dynamics of flapping flight to show a vibrational control mechanism due to the oscillation of the driving aerodynamic forces. We also experimentally demonstrate such a phenomenon on a flapping apparatus that has two degrees of freedom: forward translation and pitching motion. It is found that the time-periodic dynamics of the flapping micro-air-vehicle is naturally (without feedback) stabilized beyond a certain threshold. Moreover, if the averaged aerodynamic thrust force is produced by a propeller revolving at a constant speed while maintaining the wings stationary at their mean positions, no stabilization is observed. Hence, it is concluded that the observed stabilization in the flapping system at high frequencies is due to the oscillation of the driving aerodynamic force and, as such, flapping flight indeed enjoys vibrational stabilization.