The effect of structural deformation on flapping wing energetics

Abstract Both flapping insect and robotic wings undergo structural deformation during flight. Deformation gives rise to considerable strain energy storage, and this energy can be recovered to accelerate/decelerate the wing over a flapping cycle. This mechanism is believed to reduce the inertial power requirements of flight. To investigate this, we develop a reduced-order flapping wing model integrating flexible structural mechanics and rigid aerodynamics. The model is applicable only for wings that experience small deformation with respect to prescribed rigid body motion. We apply the model to a simplified hawkmoth Manduca sexta forewing. The idealized wing is first assumed to undergo single degree-of-freedom rotation in vacuum. We determine that a tuned compliant wing requires 20% less RMS mechanical power than a rigid flapping wing with identical kinematics. The model is then applied to a wing undergoing three-dimensional rotation in air, and optimal flapping kinematics and wing fundamental frequency are determined. Optimized rotation amplitudes and fundamental frequency agree within reason to biologically observed values with the largest discrepancy occurring in pitching amplitude. The optimal flapping-to-fundamental frequency ratio is ω ∕ ω 1  ≈ 0.35, though compliance reduces overall energetic expenditures over a wide range of frequency ratios. Under the optimal configuration, compliance reduces RMS power by 25% compared to a rigid wing, suggesting wing flexibility plays a significant role in insect flight efficiency. These results highlight the necessity for modal testing of robotic flapping wings. The natural frequency of the wing can be tuned by varying the effective diameter of the wing strut such that maximal efficiency is achieved. Tuned elastic wings can be used with other compliant drivetrain components to maximize energy efficiency in flapping-wing robots.