Optimal hovering kinematics with respect to various flapping-wing shapes

Flapping-wing kinematics for insect flight has been studied for decades, especially since engineers became interested in flapping-wing micro air vehicles (FWMAVs). Previous work mainly focused on understanding kinematic patterns employed by different insects from the perspective of their trajectories and aerodynamics, and on optimization of kinematic parameters to enhance the understanding. However, systematic research on the impact of different wing shapes and corresponding kinematics is incomplete. In this paper, we search for energy-efficient kinematics for hovering flight for a series of wing shapes which are described by a Beta probability density function and the shape parameter r̂1 (the non-dimensional radius of the first moment of wing area) to guide the wing design for FWMAVs. Three kinematic patterns are considered in the optimization: (1) fully active and harmonic kinematics for rigid wings, (2) active kinematics with linear torsion from the wing root to wing tip, (3) kinematics with passive pitching motion. We found that for the first kinematic pattern more efficient hovering flight can be achieved by the wing shape with a larger r̂1, a smaller frequency and no heaving motion, and linearly torsional pitching leads to more energy-efficient flight compared with kinematics with constant pitching amplitude along the leading edge. Additionally, optimal hovering kinematics with passive pitching is accompanied by heaving motion irrespective of its wing shape, which is reflected in insect flight. Although it is important to generate sufficient lift force for hovering with passive kinematics, the presence of the heaving motion dramatically increases the energy consumption.