An Analytical Investigation of Flapping Wing Structures for Micro Air Vehicles

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An analytical model of flapping wing structures for bio-inspired micro air vehicles is presented in this dissertation. Bio-inspired micro air vehicles (MAVs) are based on insects and hummingbirds. These animals have lightweight, flexible wings that undergo large deformations while flapping. Engineering studies have confirmed that deformations can increase the lift of flapping wings. Wing flexibility has been studied through experimental construction-and-evaluation methods and through computational numerical models. Between experimental and numerical methods there is a need for a simple method to model and evaluate the structural dynamics of flexible flapping wings. This dissertation's analytical model addresses this need.

A time-periodic assumed-modes beam analysis of a flapping, flexible wing undergoing linear deformations is developed from a beam analysis of a helicopter blade. The resultant structural model includes bending and torsion degrees of freedom. The model is non-dimensionalized. The ratio of the system's structural natural frequency to wingbeat frequency characterizes its constant stiffness, and the amplitude of flapping motion characterizes its time-periodic stiffness. Current flapping mechanisms and MAVs are compared to biological fliers on the basis of the characteristic parameters. The beam analysis is extended to develop an plate model of a flapping wing.

The time-periodic stability of the flapping wing model is assessed with Floquet analysis. A flapping-wing stability diagram is developed as a function of the characteristic parameters. The analysis indicates that time-periodic instabilities are more likely for large-amplitude, high-frequency flapping motion. Instabilities associated with the first bending mode dominate the stability diagram. Due to current limitations of flapping mechanisms, instabilities are not likely in current experiments but become more likely at the operating conditions of biological fliers.

The effect of structural design parameters, including wing planform and material stiffness, are assessed with an assumed-modes aeroelastic model. Wing planforms are developed from an empirical model of biological planforms. Non-linearities are described in the effect of membrane thickness on lift generation. Structural couplings due to time-periodic stiffness are identified that can decrease lift generation at certain wingbeat frequencies.