Summary: We have constructed equations of motion for an ornithopter with periodic wing motions. The aerodynamics are a function of the vehicle states, and the physics reveals complex nonlinear relationships for those states. Proposed was a root solving method which used limit cycles that characterized the periodic motion in an ornithopter's states without a priori estimates of the vehicle's trim velocities, or even proof of their existence.

Limit cycles are merely a feature of the phase space in ornithopter flight dynamics, that when found herald powerful implications. They represent the only analog to a dynamic equilibrium in this timeexplicit (nonautonomous) dynamic system; with flapping wings, there is no constant force and only a trivial case would maintain constant states in the system. We found a numerical approximation of a limit cycle to computer precision allowing for a study of the system's stability with the same precision as calculating the eigenvalues of the Jacobian of an equilibrium point. In our analysis, the estimation of the system's eigenvalues is independent of the flapping frequency and amplitude, which figure largely in averaging methods. The least precise part of the dynamic analysis is now the aerodynamic model. The Floquet algorithm provides a starting point to find other trim points (different flapping frequencies, wing twist amplitudes, or elevator deflections), necessary for a more comprehensive vehicle design. Future possibilities are to use this methodology as a basis to solve for optimal kinematics for lowpower trajectories, high-speed trajectories, and unsteady maneuvers such as takeoff, landing, and lateral movements.

Abstract: Energy harvesters have gained much attention as renewable energy source applications within wireless sensor technology. Focus has been directed mostly in two realms, maximizing energy output and efficient conversion via energy management circuitry. More analysis is still needed though on the fundamentals of operation in order to optimize for the size and amount of piezoelectric material needed for energy harvester applications. This work extends on the modeling of piezoelectric cantilevers by adding in the geometry of variable cross-sections, exploring standard rectangular designs and configurations with tapers and curvatures. By changing the geometry, a change in the beam strain profile is induced and thus a change in the voltage output. Experimental results are included to show actual performance outputs of each of the designs.

Keywords: Energy harvesting, piezoelectrics, modeling, variable geometry, variable cross-sectional area

Abstract: The quasi-steady aerodynamics model is coupled to a dynamic model of ornithopter flight. Previously, the combined model has been used to calculate forward flight trajectories, each a limit cycle in the vehicle's states. The limit cycle results from the periodic wing beat, producing a periodic force while on the cycle's trajectory. This was accomplished using a multiple shooting algorithm and numerical integration in MATLAB. An analysis of hover, a crucial element to vertical takeoff and landing in adverse conditions, follows. A method to calculate plausible wing flapping motions and control surface deflections for hover is developed, employing the above flight dynamics model. Once a hovering limit cycle trajectory is found, it can be linearized in discrete time and analyzed for stability (by calculating the trajectory's Floquet multipliers a type of discrete-time eigenvalue) are calculated. The dynamic mode shapes are discussed.

Keywords: Ornithopter, stability, trim, hovering flight, Floquet

Abstract: The quasi-steady aerodynamics model and the vehicle dynamics model of ornithopter flight are explained, and numerical methods are described to capture limit cycle behavior in ornithopter flight. The Floquet method is used to determine stability in forward flight, and a linear discrete-time state-space model is developed. This is used to calculate stabilizing and disturbance-rejecting controllers.

Keywords: Ornithopter, Monodromy, Floquet multipliers

Abstract: The quasi-steady aerodynamics model and the vehicle dynamics model of ornithopter flight are explained, and numerical methods are described to capture limit cycle behavior in ornithopter flight. The Floquet method is used to determine stability in forward flight, and a linear discrete-time state-space model is developed. This is used to calculate stabilizing and disturbance-rejecting controllers.

Keywords: Ornithopter, Monodromy, Floquet multipliers

Abstract: A quasi-steady model of flapping-wing aerodynamics developed for falling cards and insect flight is adapted for ornithopter design. The time-variant force calculations are coupled with the standard vehicle dynamics equations. Trim is defined for vehicles with periodic forcing as a condition of periodic state trajectories (with the same period). An optimization method is used to find wing motion parameters and initial conditions producing trim conditions at a specified average forward speed and elevation, while minimizing power consumption.

Abstract: A quasi-steady model of flapping-wing aerodynamics developed for falling cards and insect flight is adapted for ornithopter design. The time-variant force calculations are coupled with the standard vehicle dynamics equations. Trim is defined for vehicles with periodic forcing as a condition of periodic state trajectories (with the same period). An optimization method is used to find wing motion parameters and initial conditions producing trim conditions at a specified average forward speed and elevation, while minimizing power consumption. The results show that separate degrees of freedom are necessary for twisting and heaving motions to yield acceptable flight conditions.

Keywords: Flapping-Flight, Design, Power, Kinematics, Ornithopter