Geometric Formulation for the Dynamics of Monarch Butterfly with the Effects of Abdomen Undulation

摘要

This paper presents a geometric formulation of the dynamics of a flapping wing aerial vehicle and utilize it to study the flight dynamics of a Monarch butterfly. The proposed model is essentially articulated rigid bodies, where two wings and an abdomen are connected to a thorax via spherical joint. An intrinsic form of Lagrangian mechanics is developed to study the inertial effects of the relative rotation between each part. Next, a quasi-steady aerodynamic model is presented without relying on the common assumption that the flapping frequency is sufficiently large. Consequently, it is suitable to study the flight of a butterfly that is characterized by relatively large wings flapping in a lower frequency. The outcome of the proposed model is compared with the captured motion of a live Monarch butterfly. It is shown that the undulation of the abdomen increases the climb rate and the forward velocity, and the motion of Monarch butterfly yields a stable period orbit.