Peter J. Attar

A comparison of classical and high dimensional harmonic balance approaches for a Duffing oscillator

The present study focuses on a novel harmonic balance formulation, which is much easier to implement than the standard/classical harmonic balance method for complex nonlinear mathematical models and algorithms. Both harmonic balance approaches are applied to Duffings oscillator to demonstrate the advantages and disadvantages of the two approaches. A fundamental understanding of the difference between these two methods is achieved, and the properties of each method are analyzed in detail.

A theoretical and experimental investigation of the effects of a steady angle of attack on the nonlinear flutter of a delta wing plate model

Limit cycle oscillations (LCO) of wings on certain modern high performance aircraft have been observed in flight and in wind tunnel experiments. Whether the physical mechanism that gives rise to this behavior is a fluid or structural nonlinearity or both is still uncertain. It has been shown that an aeroelastic theoretical model with only a structural nonlinearity can predict accurately the limit cycle behavior at low subsonic flow for a plate-like wing at zero angle of attack. Changes in the limit cycle and flutter behavior as the angle of attack is varied have also been observed in flight. It has been suggested that this sensitivity to angle of attack is due to a fluid nonlinearity. In this investigation, we study the flutter and limit cycle behavior of a wing in low subsonic flow at small steady angles of attack. Experimental results are compared to those predicted using an aeroelastic theoretical model with only a structural nonlinearity. Results from both experiment and theory show a change in flutter speed as the steady angle of attack is varied. Also the LCO magnitude increased at a given velocity as the angle of attack was increased for both the experiment and theory. While not proving that the observed sensitivity to angle of attack of LCO in aircraft is due to a structural nonlinearity, the results do show that a change in the aeroelastic behavior at angles of attack can be caused by a structural nonlinearity as well as a fluid nonlinearity. In this paper, only structural nonlinearities are considered, but an extension to include aerodynamic nonlinearities would be very worthwhile.

Modeling Delta Wing Limit-Cycle Oscillations Using a High-Fidelity Structural Model

Flutter and limit-cycle oscillations(LCO) of a delta-wing model are studied theoretically and correlated with results from an earlier experiment and an earlier simpler theoretical model. The present theoretical model uses a high-fidelity nonlinear structural model and a linear vortex lattice aerodynamic model. The commercial finite element package ANSYS is selected to model the structure and is coupled to the vortex lattice aerodynamic model using a subiteration procedure to carry out time simulations. The delta-wing model is studied for five angles of attack (0, 1, 2, 3, and 4 deg) and for various flow speeds. Theoretical results are calculated for two different root-chord boundary conditions, that is, fully fixed and also another that allows some in-plane movement at the root chord by attaching stiff in-plane springs that connect the structure to the root boundary. The results obtained using the high-fidelity structural model are compared to earlier results computed using a lower-fidelity von Karman plate theory. For all angles of attack studied here, the correlation between theory and experiment is better for the aeroelastic model, which uses the high-fidelity (ANSYS) structural model. Both flutter velocity and frequency as well as the LCO amplitudes and frequencies that are predicted using the higher-fidelity stuctural model correlate well with experiment. In particular the flutter and LCO results predicted using the high-fidelity structural model are similar, both qualitatively and quantitatively, for the two different in-plane boundary conditions. However the results obtained from the von Karman model differ substantially for the two different in-plane boundary conditions.

Nonlinear Aeroelastic Study for Folding Wing Structures

DOI: 10.2514/1.44868 A folding wing structure consisting of three components (a fuselage, an inboard wing, and an outboard wing) is modeled computationally using a geometrically nonlinear structural dynamics theory based upon von Karman strainsandathree-dimensionalvortexlatticeaerodynamicmodelwithanexacttangent flowboundaryconditionand planar wake assumption. The structural dynamic equations of motion are discretized in space using a discrete Ritz basis derived from finite element analysis and component synthesis. Results from the computational model are comparedwiththosefromexperimentsdesignedandtestedintheDukeUniversitywindtunnelforthreefoldingwing configurations.Stablelimitcycleoscillationsat flowvelocitiesbeyondthelinear fluttervelocityaremeasuredinwindtunnel experiments and predicted using the computational model. Overall, the limit cycle oscillation magnitude and dominant response frequency results from theory show good agreement with those measured in the experiment. Qualitatively, both the experimental and theoretical limit cycle oscillation curves for the inboard wing show limited nonlinear stiffening with flow velocity for the range of velocities tested. The theoretical model also predicts that the outboard wing limit cycle oscillation tip displacements for the folding wing configuration with the largest outboard folding angle is significantly higher than the two other configurations. Unlike the inboard wing, for each configuration, the outboard wing theoretical limit cycle oscillation curves do show a trend that is reminiscent of a stiffening nonlinearity.

Implicit LES Simulations of a Low Reynolds Number Flexible Membrane Wing Airfoil

This paper presents implicit LES simulations of a flexible membrane wing airfoil at a transitional/turbulent Reynolds number. The aerodynamics are simulated using a wellvalidated, sixth-order Navier-Stokes solver which is coupled with a one-dimensional finiteelement approach for the structural dynamic response of the membrane. The membrane airfoil geometry chosen corresponds to the experimental configuration of Rojratsirikul et al. 1 Computations on a coarse and refined mesh were performed to assess the impact of grid resolution on the computed solutions. A description of the unsteady fluid/structure interaction for angles of attack of = 8 and = 14 are presented indicating a close coupling between the unsteady flow behavior and the structural response. Initial comparisons of the computational results with available experimental data show good qualitative agreement. Issues with membrane structural modeling and the need for a more complete experimental characterization of the membrane structural properties are discussed. In order to address the technical challenges associated with successful MAV development, designers are looking to biological flight for inspiration. Successful development of these biomimetic MAV concepts will require significant advancements in the fundamental understanding of the unsteady aerodynamics of low Reynolds number fliers and the associated fluid-structure interactions. The inherent flexibility in the structural design of lightweight MAVs and the exploitation of that flexibility creates strong coupling between the unsteady fluid dynamics and the airframe structural response giving rise to tightly integrated, multidisciplinary physics. Conventional simplified analytical techniques and empirical design methods, although attractive for their eciency, may have limited applicability for these complicated, multidisciplinary design problems. Critical insight into the highly complex, coupled MAV physics calls for the exploitation of advanced multidisciplinary computational techniques. The focus of the present paper will be the simulation and analysis of aeroscience issues associated with a flexible membrane wing airfoil. The specific case to be considered corresponds to the experiments of Rojratsirikul, Wang and Gursul 1 where flow visualizations as well as PIV measurements have been carried out for a simple membrane wing. An implicit LES approach 2 is employed to compute the mixed laminar/transitional/turbulent flowfields present in the experiments of Rojratsirikul et al. The ILES approach exploits the properties of a well validated, robust, sixth-order Navier-Stokes solver. 3‐5 This aerodynamic solver is coupled with a one-dimensional finite element membrane structural model suitable for the highly nonlinear structural response associated with a flexible membrane airfoil. In a previous paper by Gordnier, 6 two-dimensional computations for very low Reynolds numbers (Re < 10 4 ) were performed for the same membrane wing configuration. The impact of various fluid and structural parameters including angle of attack, membrane elasticity, membrane pretension and Reynolds number were explored. The present work will extend these computations to the Reynolds number of the experiments of Rojratsirikul et al, Re = 48,500. At this Reynolds number the flow is transitional/turbulent and

Flutter/limit cycle oscillation analysis and experiment for wing-store model

A delta wing experimental model with an external store has been designed and tested in the Duke University wind tunnel. The wing structure is modeled theoretically by using von Karman plate theory that allows for geometric strain-displacement nonlinearities in the plate wing structure. A component modal analysis is used to derive the full structural equations of motion for the wing/store combination system. A three-dimensional time domain vortex lattice aerodynamic model including a reduced-order model aerodynamic technique and a slender-body aerodynamic theory for the store are also used to investigate the nonlinear aeroelastic system. The effects of the store pitch stiffness (attachment stiffness), the span location of store, and the store aerodynamics on the critical flutter velocity and limit cycle oscillations (LCO) are discussed. The correlations between the theory and experiment are good for both the critical flutter velocity and frequency but not good for the LCO amplitude, especially when the store is located near the wing tip. The theoretical structural model needs to be improved to determine LCO response, and improved results are shown in the companion paper as obtained with a higher-order structural model.

Experimental Characterization of Limit Cycle Oscillations in Membrane Wing Micro Air Vehicles

The idea of using small-scale vehicles, often termed micro air vehicles, for various surveillance applications has become increasingly popular in recent years. A micro air vehicle design of particular interest is the membrane wing micro air vehicles, in which the structural skeleton is covered with a thin membrane instead of conventional wing skin materials, developed in particular for its lightweight nature, static stability, and passive gust rejection. In the current work, membrane wing micro air vehicles are developed and tested experimentally in order to determine the structural response of batten-reinforced membrane wing micro air vehicles to varying conditions: small angles of attack, number of battens, and membrane pretension. A self-excited instability (flutter) was noted for each model with limit cycle oscillations occurring at postflutter flow velocities. Small angles of attack had little effect on the flutter velocity, frequency, and mode for a given configuration, while increasing the membrane pretension delayed flutter and reduced the magnitude of limit cycle oscillation experienced by the model at a given flow velocity. Increasing the number of structural battens for the membrane wing micro air vehicle models also delayed the flutter velocity and reduced the magnitude of limit cycle oscillation at a given flow velocity while altering the flutter mode.

Stochastic Analysis of a Nonlinear Aeroelastic Model Using the Response Surface Method

An efficient method is presented for quantifying the effect of parametric uncertainty on the response of a nonlinear aeroelastic system. The proposal stochastic model uses a response surface method to map the random input parameters of the system to the specified system output (in this instance root-mean square wing tip response). To handle the bifurcation in the response surface due to aeroelastic self-excited instability, the response surface model is fit using a two region regression. The results from this model are compared to those from a full Monte Carlo simulation for both a one-dimensional random input parameter model (thickness) and a two-dimensional random input parameter model (thickness and modulus of elasticity). The response surface method results compare favorably with the full model results while achieving a 2 to 3 order of magnitude gain in computational efficiency.

Implicit LES Simulations of a Flexible Flapping Wing

A high-order (up to 6th order) Navier-Stokes solver is coupled with a structural solver that decomposes the equations of three-dimensional elasticity into cross-sectional, smalldeformation and spanwise, large-deformation analyses for slender wings. The resulting high-fidelity aeroelastic solver is applied to the investigation of both a rigid and moderately flexible rectangular wing undergoing a pure plunging motion. Comparisons of the computed results demonstrate good agreement with available experimental measurements. A description of the complex interaction between the unsteady aerodynamics and the flexible wing structural dynamics is given. Connections between the results of this analysis and the enhanced aerodynamic loads for the flexible wing are made.

Reduced Order Nonlinear System Identification Methodology

A new method is presented which enables the identification of a reduced order nonlinear ordinary differential equation (ODE) which can be used to model the behavior of nonlinear fluid dynamics. The method uses a harmonic balance technique and proper orthogonal decomposition to compute reduced order training datawhich is then used to compute the unknown coefficients of the nonlinear ODE. The method is used to compute the Euler compressible flow solutions for the supercritical two-dimensional NLR-7301 airfoil undergoing both small and large pitch oscillationsatthreedifferentreducedfrequenciesandataMachnumberof0.764.Steadyanddynamicliftcoefficient datacomputedusingathreeequationreducedordersystemidentificationmodelcomparedwellwithdatacomputed using the full CFD harmonic balance solution. The system identification model accurately predicted a nonlinear trend in the lift coefficient results (steady and dynamic) for pitch oscillation magnitudes greater than 1 deg. Overall the reduction in the number of nonlinear differential equations was 5 orders of magnitude which corresponded to a 3 order of magnitude reduction in the total computational time.