Natural fliers with flapping wings face the dual challenges of energy efficiency and active control of wing motion for achieving diverse modes of flight. It is hypothesized that flapping-wing systems use resonance to improve muscle mechanical output energy efficiency, a principle often followed in bioinspired flapping-wing robots. However, resonance can limit the degree of active control, a trade-off rooted in the dynamics of wing motor systems and can be potentially reflected in muscle work loops. To systematically investigate how energy efficiency trades off with active control of wingbeat frequency and amplitude, here we developed a parsimonious model of the wing motor system with either synchronous or asynchronous power muscles. We then non-dimensionalized the model and performed simulations to examine model characteristics as functions of Weis-Fogh number and dimensionless flapping frequency. For synchronous power muscles, our model predicts that energy efficiency trades off with frequency control rather than amplitude control at high Weis-Fogh numbers; however, no such trade-off was found for models with asynchronous power muscles. The work loops alone are insufficient to fully capture wing motor characteristics, and therefore fail to directly reflect the trade-offs. Finally, using simulation results, we predict that natural fliers function at Weis-Fogh numbers close to 1.
