Search for: All records

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.

 

We propose a novel 3D source seeking algorithm for rigid bodies with a non-collocated sensor inspired by the chemotactic navigation strategy of sea urchin sperm known as helical klinotaxis. We work directly with the rotation group SO(3) without parameterization in representing the attitude of a rigid body. As a consequence, the proposed algorithm does not require attitude feedback for implementation as opposed to all previous work on 3D source seeking. The stability of the proposed algorithm is proven using an intricate combination of singular perturbation and second order averaging. 

We propose a novel 3D source seeking algorithm for rigid bodies with a non-collocated sensor inspired by the chemotactic navigation strategy of sea urchin sperm known as helical klinotaxis. We work directly with the rotation group SO(3) without parameterization in representing the attitude of a rigid body. As a consequence, the proposed algorithm does not require attitude feedback for implementation as opposed to all previous work on 3D source seeking. The stability of the proposed algorithm is proven using an intricate combination of singular perturbation and second order averaging. 

Vibrational control is an open loop stabilization technique via the application of highamplitude, high-frequency oscillatory inputs. The averaging theory has been the standard technique for designing vibrational control systems. However, it stipulates too high oscillation frequency that may not be practically feasible. Therefore, although vibrational control is very robust and elegant (stabilization without feedback), it is rarely used in practical applications. The only well-known example is the Kapitza pendulum; an inverted pendulum shose pivot is subject to vertical oscillation. the unstable equilibrium of the inverted pendulum gains asymptotic stability due to the high-frequency oscillation of the pivot. In this paper, we provide a new vibrational control system from Nature; flapping flight dynamics. Flapping flight is a rich dynamical system as a representative model will typically be nonlinear, time-varying, multi-body, multi-time-scale dynamical system. Over the last two decades, using direct averaging, there has been consensus in the flapping flight dynamics community that insects are unstable at the hovering equilibrium due to the lack of pitch stiffness. In this work, we perform higher-order averaging of the time-periodic dynamics of flapping flight to show a vibrational control mechanism due to the oscillation of the driving aerodynamic forces. We also experimentally demonstrate such a phenomenon on a flapping apparatus that has two degrees of freedom: forward translation and pitching motion. It is found that the time-periodic dynamics of the flapping micro-air-vehicle is naturally (without feedback) stabilized beyond a certain threshold. Moreover, if the averaged aerodynamic thrust force is produced by a propeller revolving at a constant speed while maintaining the wings stationary at their mean positions, no stabilization is observed. Hence, it is concluded that the observed stabilization in the flapping system at high frequencies is due to the oscillation of the driving aerodynamic force and, as such, flapping flight indeed enjoys vibrational stabilization. 

Flapping-Wing Micro-Air-Vehicles (FWMAVs) are bio-inspired air vehicles that mimic insect and bird flight. The dynamic behavior of these systems is typically described by a multi-body, multi-time-scale, nonlinear, time-varying dynamical system. Interestingly, this rich dynamics lead to unconventional stabilization mechanisms whose study essentially necessitates a mathematically rigorous analysis. In this paper, we use higherorder averaging, which is based on chronological calculus, to show that insects and their man-made counterparts (FWMAVs) exploit vibrational control to stabilize their body pitching angle. Such an unconventional stabilization cannot be captured by direct averaging. We also experimentally demonstrate such a phenomenon by constructing an experimental setup that allows for two degrees of freedom for the body; forward motion and pitching motion. We measure the response of the body pitching angle using a digital camera and an image processing algorithm at different flapping frequencies. It is found that there is a flapping frequency threshold beyond which the body pitching response is naturally (without feedback) stabilized, which conforms with the vibrational control concept. Moreover, we also construct a replica of the experimental setup with the FWMAV being replaced by a propeller revolving at constant speed, which results in a constant aerodynamic force, leaving no room for vibrational control. The response of the propellersetup is unstable at all frequencies, which also corroborates the fact that the observed stabilization of the FWMAV-setup at high frequencies is a vibrational stabilization phenomenon. 

Flapping propulsion has been deemed inefficient for practical use in thrusting actual airplanes. In this paper, we revisit this claim in the light of several recent findings on the unsteady aero/hydro dynamics of natural flyers/swimmers (e.g., birds, insects, cetaceans). We propose a new airplane concept, called the Flapping-Tail Concept Airplane (FTCA), in which the horizontal tail is driven by a power shaft into a pitching-plunging-flapping motion through a flapping mechanism. For such a concept, we show that there is a significant room for boosting flapping propulsive efficiency that may outperform the current turbofan engine technologies. We use Garrick’s classical unsteady aerodynamic model for flapping propulsion to show that allowing for a simultaneous flap deflection with pitching-plunging may enhance the propulsive efficiency by 20%. Moreover, we propose other promising interacting flow mechanisms that may enhance the propulsive efficiency even more and provide a geometric control theoretic formulation to guide such an interaction. We also show the favorable effect of operating in the stall regime with large amplitudes. Finally, we study the effect of such an oscillating tail on the flight mechanics of the airplane and provide recommendations for future investigations necessary to make the proposed vision come closer to real applications.