Linear Time-Invariant Models of the Dynamics of Flapping-Wing Flight

This paper demonstrates the extension of the harmonic decomposition methodology, originally developed for rotorcraft applications, to the study of the nonlinear time-periodic dynamics of flapping-wing flight. A harmonic balance algorithm based on harmonic decomposition is successfully applied to find the periodic equilibrium and approximate linear time-invariant dynamics about that equilibrium of the vertical and longitudinal dynamics of a hawk moth. These approximate linearized models are validated through simulations against the original nonlinear time-periodic dynamics. Dynamic stability using the linear models is assessed and compared to that predicted using the averaged dynamics. In addition, modal participation factors are computed to quantify the influence of the higher harmonics on the flight dynamic modes of motion. The study shows that higher harmonics play a key role in the overall dynamics of f lapping-wing flight. The higher harmonics are shown to induce a vibrational stabilization mechanism that increases the pitch damping and stiffness while reducing the speed stability. This mechanism results in the stabilization of the pitch oscillation mode and thus of the longitudinal hovering cubic. As such, the findings of this study suggest that if a hovering vehicle is excited by periodic forcing at sufficiently high frequency and amplitude, its hovering flight dynamics may become stable.