Resonance principle for the design of flapping wing micro air vehicles

Achieving resonance in flapping wings has been recognized as one of the most important principles to enhance power efficiency, lift generation, and flight control performance of high-frequency flapping wing micro air vehicles (MAVs). Most of work on the development of such vehicles have attempted to achieve wing flapping resonance. However, the theoretical understanding of its effects on the response and energetics of flapping motion has lagged behind, leading to sub-optimal design decisions and misinterpretations of experimental results. In this work, we systematically model the dynamics of flapping wing as a forced nonlinear resonant system. Using linear approximation approach, we derived analytic solution for steady-state flapping amplitude, energetics, and characteristic frequencies including natural frequency, damped natural frequency, and peak frequency. Our results showed that both aerodynamic lift and power efficiency are maximized by driving the wing at natural frequency, instead of other frequencies. Interestingly, the flapping velocity is maximized at natural frequency as well, which can lead to an easy experimental approach to identify natural frequency and validate the resonance design. Our models and analysis were validated with both simulations and experiments on ten different wings mounted a direct-motor-drive flapping wing MAV. The result can serve as a systematic design principle and guidance in the interpretations of empirical results.