Using a dynamically scaled robotic wing, we studied the aerodynamic torque generation of flapping wings during roll, pitch, and yaw rotations of the stroke plane. The total torque generated by a wing pair with symmetrical motions was previously known as flapping counter-torques (FCTs). For all three types of rotation, stroke-averaged FCTs act opposite to the directions of rotation and are collinear with the rotational axes. Experimental results indicate that the magnitude of FCTs is linearly dependent on both the flapping frequency and the angular velocity. We also compared the results with predictions by a mathematical model based on quasi-steady analyses, where we show that FCTs can be described through consideration of the asymmetries of wing velocity and the effective angle of attack caused by each type of rotation. For roll and yaw rotations, our model provided close estimations of the measured values. However, for pitch rotation the model tends to underestimate the magnitude of FCT, which might result from the effect of the neglected aerodynamics, especially the wake capture. Similar to the FCT, which is induced by body rotation, we further provide a mathematical model for the counter force induced by body translation, which is termed as flapping counter-force (FCF). Based on the FCT and FCF models, we are able to provide analytical estimations of stability derivatives and to study the flight dynamics at hovering. Using fruit fly (Drosophila) morphological data, we calculated the system matrix of the linearized flight dynamics. Similar to previous studies, the longitudinal dynamics consist of two stable subsidence modes with fast and slow time constants, as well as an unstable oscillatory mode. The longitudinal instability is mainly caused by the FCF induced by an initial forward/backward velocity, which imparts a pitch torque to the same direction of initial pitch velocity. Similarly, the lateral dynamics also consist of two stable subsidence modes and an unstable oscillatory mode. The lateral instability is mainly caused by the FCF induced by an initial lateral velocity, which imparts a roll torque to the same direction of initial roll velocity. In summary, our models provide the first analytical approximation of the six-degree-of-freedom flight dynamics, which is important in both studying the control strategies of the flying insects and designing the controller of the future flapping-wing micro air vehicles (MAVs).
Mathematical Modelling of Near-Hover Insect Flight Dynamics
