Flapping insect wings deform under aerodynamic as well as inertial-elastic forces. This deformation is thought to improve power economy and reduce the energetic costs of flight. However, many flapping wing models employ rigid body simplifications or demand excessive computational power, and are consequently unable to identify the influence of flexibility on flight energetics. Here, we derive a reduced-order model capable of estimating the driving torques and corresponding power of flapping, flexible insect wings. We validate this model by actuating a tobacco hornworm hawkmoth Manduca sexta (L.) forewing with a custom single-degree-of-freedom mechanical flapper. Our model predicts measured torques and instantaneous power with reasonable accuracy. Moreover, the flexible wing model predicts experimental trends that rigid body models cannot, which suggests compliance should not be neglected when considering flight dynamics at this scale. Next, we use our model to investigate flight energetics with realistic flapping kinematics. We find that when the natural frequency of the wing is roughly three times that of the flapping frequency, flexibility can reduce energy expenditures by almost 25% compared to a rigid wing if negative work is stored as potential energy and subsequently released to do positive work. The wing itself can store about 30% of the 1200 µJ of total energy required over a wingbeat. Peak potential energy storage occurs immediately before stroke reversal. We estimate that for a moth weighing 1.5-2.5 g, the peak instantaneous power required for flight is 75-125 W kg −1 . However, these peak values are likely lower in natural insect flight, where the wing is able to exchange strain energy with the compliant thorax. Our findings highlight the importance of flexibility in flapping wing micro aerial vehicle design and suggest tuned flexibility can greatly improve vehicle efficiency.
Fluid–structure interaction (FSI) plays a significant role in the deformation of flapping insect wings. However, many current FSI models are high-order and rely on direct computational methods, thereby limiting parametric studies as well as insights into the physics governing wing dynamics. We develop a novel flapping wing FSI framework that accommodates general wing geometry and fluid loading. We use this framework to study the unilaterally coupled FSI of an idealized hawkmoth forewing considering two fluid models: Reynolds-averaged Navier–Stokes computational fluid dynamics (RANS CFD) and blade element theory (BET). We first compare aerodynamic modal forces estimated by the low-order BET model to those calculated via high fidelity RANS CFD. We find that for realistic flapping kinematics, BET estimates modal forces five orders of magnitude faster than CFD within reasonable accuracy. Over the range flapping kinematics considered, BET and CFD estimated modal forces vary maximally by 350% in magnitude and approximately π/2 radians in phase. The large reduction in computational time offered by BET facilitates high-dimensional parametric design of flapping-wing-based technologies. Next, we compare the contributions of aerodynamic and inertial forces to wing deformation. Under the unilateral coupling assumption, aerodynamic and inertial-elastic forces are on the same order of magnitude—however, inertial-elastic forces primarily excite the wing’s bending mode whereas aerodynamic forces primarily excite the wing’s torsional mode. This suggests that, via conscientious sensor placement and orientation, biological wings may be able to sense independently inertial and aerodynamic forces.
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