Wing inertia as a cause of aerodynamically uneconomical flight with high angles-of-attack in hovering insects

Flying insects can maintain maneuverability in the air by flapping their wings, and to save energy, the wings should operate following the optimal kinematics. However, unlike conventional rotary wings, insects operate their wings at aerodynamically uneconomical and high angles-of-attack (AoAs). Although insects have continuously received attention from biologists and aerodynamicists, the high AoA operation in insect flight has not been clearly explained. Here, we use a theoretical blade-element model to examine the impact of wing inertia on the power requirement and flapping AoA, based on three-dimensional free-hovering flight wing kinematics of a horned beetle, The relative simplicity of the model allows us to search for the best AoAs distributed along the wingspan, which generate the highest vertical force per unit power. We show that, although elastic elements may be involved in flight muscles to store and save energy, the insect still has to spend substantial power to accelerate its wings, because inertial energy stores should be used to overcome aerodynamic drag before being stored elastically. At the same flapping speed, a wing operating at a higher AoA requires lower inertial torque, and therefore lower inertial power output, at the stroke reversals than a wing operating at an aerodynamically-optimal low AoA. An interactive aerodynamic-inertial effect thereby enables the wing to flap at sufficiently high AoAs, which causes an aerodynamically uneconomical flight in an effort to minimize the net flight energy.