Revisiting resonance in flapping flight: Supra-resonant oscillations in moths, bees, and robophysical models

Most insects fly with indirect actuation using muscles acting in parallel to elastic exoskeletons and coupled in series to flapping wings. This combination results in resonant dynamics even if it differs from classic viscoelastic oscillators because of the presence of structural (frequency-independent) damping and aerodynamic drag. Moreover, muscle force can periodically drive wingstrokes (termed synchronous flight) or incorporate stretch activation resulting in self-excited, limit-cycle oscillations decouple from neural pacing (asynchrony) or be a combination of the two. Yet while insects have resonant mechanics, do they operate at their resonant frequency? And what are the different implications for periodically forced and self-excited flapping flight especially when both types of forcing are present? Here we show that synchronous hawkmoths and asynchronous bumblebees are both supra-resonant, operating above their kinematic resonance peak. We explore these properties in a dynamically scaled spring-wing robotic flapper whose kinematics are emergent. Spring-wing systems with both types of forcing can transition from one mode of flight to the other across a classic entrainment boundary. Operating off-resonance does not preclude useful energy exchange but sets up a trade-off of power reduction and control via frequency modulation. We capture this tradeoff in the nondimensional Weis-Fogh number, n, the ratio of peak inertial to peak aerodynamic force, which is 1-10 for nearly all insects.