Modeling of Instantaneous Passive Pitch of Flexible Flapping Wings

While the biological flyers often showcase high deformed wing structures, the effects of flexibility on the flapping wing aerodynamics remain inadequately understood. A major challenge in the study of flexible flapping wings is that the resulting wing motion is an outcome of a dynamic balance between the structural dynamics of the wing and the unsteady aerodynamics. Since the effective angle of attack is affected by structural deformation, the wing resulting kinematics is a priori unknown. In this study, we solve for the temporal evolution of the wing deformation by modeling the wing as an elastic beam under an imposed nominal kinematics. We use the Morison equation, which consists of the added mass and aerodynamic damping forces, for the fluid dynamic force term. The resulting wing deformation is correlated with a numerical solution of fully coupled aeroelastic framework. While the end-of-the-stroke passive pitch angle agrees well with the numerical solution, the midstroke pitch angle needs to be empirically corrected. Based on these two angles, we construct a first-order harmonic passive pitch motion. The derived model can satisfactorily predict the phase lag and reasonably the angular amplitude. The current results can be a stepping stone toward formulation of a quick predictive model for instantaneous aerodynamic forces on a flexible flapping wing, for bio-flight and human engineered MicroAir Vehicles.