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Geometric control theory is the application of differential geometry to the study of nonlinear dynamical systems. This control theory permits an analytical study of nonlinear interactions between control inputs, such as symmetry breaking or force and motion generation in unactuated directions. This paper studies the unsteady aerodynamics of a harmonically pitching–plunging airfoil in a geometric control framework. The problem is formulated using the Beddoes–Leishman model, a semi-empirical state space model that characterizes the unsteady lift and drag forces of a two-dimensional airfoil. In combination with the averaging theorem, the application of a geometric control formulation to the problem enables an analytical study of the nonlinear dynamics behind the unsteady aerodynamic forces. The results show lift enhancement when oscillating near stall and thrust generation in the post-stall flight regime, with the magnitude of these force generation mechanisms depending on the parameters of motion. These findings demonstrate the potential of geometric control theory as a heuristic tool for the identification and discovery of unconventional phenomena in unsteady flows. 

In recent years, there has been a growing interest in low-Reynolds-number, unsteady flight applications, leading to renewed scrutiny of the Kutta condition. As an alternative, various methods have been proposed, including the combination of potential flow with the triple-deck boundary layer theory to introduce a viscous correction for Theodorsen's unsteady lift. In this research article, we present a dynamical system approach for the total circulatory unsteady lift generation of a pitching airfoil. The system's input is the pitching angle, and the output is the total circulatory lift. By employing the triple-deck boundary layer theory instead of the Kutta condition, a new nonlinearity in the system emerges, necessitating further investigation to understand its impact on the unsteady lift model. To achieve this, we utilize the describing function method to represent the frequency response of the total circulatory lift. Our analysis focuses on a pitching flat plate about the mid-chord point. The results demonstrate that there is an additional phase lag due to viscous effects, which increase as the reduced frequency increases, the Reynolds number decreases, and/or the pitching amplitude increases.