Translational and rotational damping of flapping flight and its dynamics and stability at hovering

Body movements of flying insects change their effective wing kinematics and, therefore, influence aerodynamic force and torque production. It was found that substantial aerodynamic damping is produced by flapping wings through a passive mechanism termed flapping countertorque during fast yaw turns. We expand this study to include the aerodynamic damping that is produced by flapping wings during body translations and rotations with respect to all its six principal axesroll, pitch, yaw, forward/backward, sideways, and heave. Analytical models were derived by the use of a quasi-steady aerodynamic model and blade-element analysis by the incorporation of the effective changes of wing kinematics that are caused by body motion. We found that aerodynamic damping, in all these cases, is linearly dependent on the body translational and angular velocities and increases with wing-stroke amplitude and frequency. Based on these analytical models, we calculated the stability derivatives that are associated with the linearized flight dynamics at hover and derived a complete 6-degree-of-freedom (6-DOF) dynamic model. The model was then used to estimate the flight dynamics and stability of four different species of flying insects as case studies. The analytical model that is developed in this paper is important to study the flight dynamics and passive stability of flying animals, as well as to develop flapping-wing micro air vehicles (MAVs) with stable and maneuverable flight, which is achieved through passive dynamic stability and active flight control.