The modeling and numerical solution for flapping wing hovering wingbeat dynamics

Abstract

The production of wingbeat motion of flapping wing hovering flight are determined by the actuating, aerodynamic and inertia forces/moments, which influence the dynamic unsteadiness and controllability of flapping wing flying. This paper presents the feasible solution for cracking the problem of two degrees of freedom (two DoFs, namely, flapping and pitch motion, respectively) highly coupled nonlinear hovering wingbeat dynamics. Firstly, two DoFs nonlinear hovering wingbeat dynamic ordinary differential equations (ODEs) are derived on basis of the extended quasi-steady aerodynamic and inertial forces/moments model. Then, we perform their numerical solution by using tractable ODEs numerical algorithm, boundary value problem-solving format, and least square method. The numerical results have a good consistency with those measured by Dr. Muijres. Moreover, the adjustable rule of phase offset of wing pitch angle relative to the flapping angle is quantificationally studied by introducing frequency ratio between pitch frequency and flapping frequency. We find that the phase offset can be directly regulated by wing pitch hinge stiffness or indirectly modulated by frequency ratio, and the peak value of wing pitch angle monotonously decreases with the increase of wing pitch hinge stiffness, opposite to the angle of attack (AoA). This adjustable rule paves a useful way for the bio-inspired flapping wing micro aerial vehicle (FWMAV) featuring passive or semi-passive pitch flexible hinge to maintain high variable AoA.