Nonlinear aeroelastic analysis of high aspect-ratio wings using the method of numerical continuation

This research explores the impact of kinematic structural nonlinearities on the dynamics of a highly deformable cantilevered wing. Two different theoretical formulations are presented and analysed for nonlinear behavior. The first formulation, which is more conventional, assumes zero equilibrias and structural nonlinearities occur as terms up to third order in the Taylor series expansion of structural nonlinearities. In the second approach, no prior assumption about equilibria states of the wing is made. Kinematic nonlinearities due to curvature and inertia were retained in their exact form. Thus, the former becomes a special case of the latter. This nonlinear formulation permits the analysis of dynamics about nonzero trims. Nonzero trim states are computed as a system parameter is varied using a continuation software tool. The stability characteristics of these trim states are also ascertained. Various bifurcation points of the system are determined. Limit-cycle oscillations are also investigated for and are characterized in terms of amplitude of vibration. The research in particular examines the impact of in-plane degree of freedom on the stability of nonzero trim states. The effect of variation of system parameters such as stiffness ratio, aspect ratio and root angle of attack is also studied. The method of direct eigenanalysis of nonzero equilibria is novel and new for an aeroelastic system

Optimal state-space reconstruction using derivatives on projected manifold

A paradigm for optimal state-space reconstruction with nonuniform time delays is proposed. A comparison based on a diffeomorphic measure and a smoothness cost function shows that the proposed methodology achieves a better reconstruction compared to a reconstruction based on time delays that are multiples of the first minimum of mutual information. It is also shown how the proposed methodology is a more reliable approach to determining the embedding dimension.8 page(s

Sparse model from optimal nonuniform embedding of time series

An approach to obtaining a parsimonious polynomial model from time series is proposed. An optimal minimal nonuniform time series embedding schema is used to obtain a time delay kernel. This scheme recursively optimizes an objective functional that eliminates a maximum number of false nearest neighbors between successive state space reconstruction cycles. A polynomial basis is then constructed from this time delay kernel. A sparse model from this polynomial basis is obtained by solving a regularized least squares problem. The constraint satisfaction problem is made computationally tractable by keeping the ratio between the number of constraints to the number of variables small by using fewer samples spanning all regions of the reconstructed state space. This helps the structure selection process from an exponentially large combinatorial search space. A forward stagewise algorithm is then used for fast discovery of the optimization path. Results are presented for the Mackey-Glass system.9 page(s

High resolution mapping of the dynamics of a nonlinear semiconductor laser system

Optical feedback is known to cause a range of complex dynamical states in the output power of semiconductor lasers. These types of systems have been much studied [1]. The dynamic state of the laser output can be controlled by varying the level of optical feedback and also the injection current to the laser [2]. Traditionally, analysis of nonlinear dynamics produced by semiconductor lasers has been based on optical and/or RF spectra, due to the high frequencies involved. More recently, the availability of high bandwidth real-time oscilloscopes has facilitated direct measurement of the output power time series and allowed the temporal information, missing from earlier investigations, to be captured. Computer controlled experimental setups have also improved the resolution at which system parameters can be varied and the amount of data that can be captured.1 page(s

Mapping transients in the nonlinear dynamics of an optically injected VCSEL

Optical injection is one of the key methods for invoking nonlinear dynamical outputs in laser systems. The system parameters that are used to control the nature of the output from such a system are the injection strength and the frequency detuning between the optical frequency of the free running master and slave lasers. A map of the dynamics can be generated using a number of measurands to facilitate identifying the fundamentally different dynamical regions in the injection-strength/frequency-detuning parameter space. Herein we describe a set of automated algorithms used to establish several measures to identify transients and instabilities in the nonlinear dynamical output of an optically injected vertical cavity surface emitting laser (VCSEL), for set and unchanging driving parameters.6 page(s

Suppressing aeroelastic instability by means of broadband targeted energy transfers, part 2: Experiments

This paper presents experimental results corroborating the analysis developed in the companion paper, Part I (Lee, Y., Vakakis, A., Bergman, L., McFarland, M., and Kerschen G., "Suppression Aeroelastic Instability Using Broadband Passive Targeted Energy Transfers, Part 1: Theory," AIAA Journal, Vol. 45, No. 3, 2007, pp. 693-711), and demonstrates that a nonlinear energy sink can improve the stability of an aeroelastic system. The nonlinear energy sink was, in this case, attached to the heave (plunge) degree of freedom of a rigid airfoil which was supported in a low-speed wind tunnel by nonlinear springs separately adjustable in heave and pitch. This airfoil was found to exhibit a at flow speeds above the critical ('flutter") speed of 9.5 m/s, easily triggered by an initial heave displacement. After attachment of a single degree of freedom, essentially nonlinear energy sink to the wing, the combined system exhibited improved dynamic response as measured by the reduction or elimination of limit cycle oscillation at flow speeds significantly greater than the wing's critical speed. The design, application, and performance of the nonlinear energy sink are described herein, and the results obtained are compared to analytical predictions. The physics of the interaction of the sink with the wing is examined in detail