Analytical Modeling for Flapping Wing Deformation and Kinematics with Beam Flexibility

The intrinsic flexibility of biological flyers' wings has been well recognized as an efficient way to regulate the performance of their flight through the complex interaction between the surrounding air and the deformation of the wings. However, for vehicles that mimic flying creatures through a flapping protocol, the deformation of the beams on the flapping wings is largely ignored. In this study, based on the elastokinetics method, theoretical models, including a simplified analytical solution, are built to determine the deformation and the kinematics of the flapping wing with beam flexibility. After verifying the models by finite element method simulations and experiments, the flapping kinematics of the wing under various wing conditions is resolved in detail. The analytical theory indicates that the system is fully regulated by two independent parameters. The first one describes the damping effect, which is the product of the driving amplitude and the damping ratio, whereas the second one describes the flexibility of the beam, which is the ratio of the driving angular velocity and the circular natural frequency of the beam. This simple analytical theory neatly captures the kinematics of the wing, which will help to evaluate the performance of the flapping wing with beam flexibility.