Modeling the Nonlinear Structural Dynamics of a Plunging Membrane Airfoil using a High Dimensional Harmonic Balance Approach

High fidelity computational models are developed to study the nonlinear structural dynamics of a plunging membrane airfoil, which is based upon previous experimental work. Two airfoil structures are investigated: a one-dimensional string and a three-dimensional Feeler gauge. Time-periodic flapping is assumed for both cases. The nonlinear equations of motion are semi-discretized in space using the finite element method, and then discretized in time using both a high dimensional harmonic balance approach and a standard finite differencing scheme. Solutions for both time-discretization methods are compared. For weak to moderately nonlinear string problems, the harmonic balance solution compares favorably with the finite difference solution, and is two to three orders of magnitude faster to compute. When the string exhibits stronger nonlinear behavior, the harmonic balance method yields results that diverge from the true solution when multiple harmonics are used. This is likely due to the presence of multiple subharmonics in the frequency response curve. The Feeler gauge problem is investigated to explore the possibility of unstable planar motion and chaotic vibrations during plunging. While the spectral contents of the response contain multiple frequencies, neither unstable planar dynamics nor chaotic vibrations were encountered within the range of parameters tested. Further investigation is needed to determine whether such a response is possible for the current model.

[1]  P. Orkwis,et al.  Adaptive harmonic balance method for nonlinear time-periodic flows , 2004 .

[2]  M. B. Rubin,et al.  NUMERICAL SOLUTIONS OF FORCED VIBRATION AND WHIRLING OF A NON-LINEAR STRING USING THE THEORY OF A COSSERAT POINT , 1996 .

[3]  Ismail H. Tuncer,et al.  A computational study on the dynamic stall of a flapping airfoil , 1998 .

[4]  Ismet Gursul,et al.  Unsteady Aerodynamics of Membrane Airfoils , 2008 .

[5]  T. Weis-Fogh.  Quick estimates of flight fitness in hovering animals , 1973 .

[6]  J. K. Bhattacharjee,et al.  Limit Cycle Oscillations , 2008 .

[7]  M. Lighthill.  Hydromechanics of Aquatic Animal Propulsion , 1969 .

[8]  Arthur W. Leissa,et al.  Large Amplitude Vibrations of Strings , 1994 .

[9]  Jeffrey P. Thomas,et al.  Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique , 2002 .

[10]  G. Carrier,et al.  A note on the vibrating string , 1949 .

[11]  Earl H. Dowell,et al.  A comparison of classical and high dimensional harmonic balance approaches for a Duffing oscillator , 2006, J. Comput. Phys..

[12]  Nicholas B. Tufillaro,et al.  An experimental investigation into the dynamics of a string , 2004 .

[13]  Peter J. Attar,et al.  High Fidelity Computational Aeroelastic Analysis of a Plunging Membrane Airfoil , 2009 .

[14]  Sam Heathcote,et al.  Effect of Spanwise Flexibility on Flapping Wing Propulsion , 2006 .

[15]  Philip Holmes,et al.  Non-linear, non-planar and non-periodic vibrations of a string , 1992 .

[16]  R Waszak Martin,et al.  Stability and Control Properties of an Aeroelastic Fixed Wing Micro Aerial Vehicle , 2001 .

[17]  Wei Shyy,et al.  Numerical Simulations of Membrane Wing Aerodynamics for Micro Air Vehicle Applications , 2005 .

[18]  Jeffrey P. Thomas,et al.  Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter, and Limit-Cycle Oscillations , 2001 .

[19]  Joanna Szmelter,et al.  A study of flow arising from insect wing flapping motion , 2002 .

[20]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[21]  Thomas J. Mueller,et al.  Thrust and Drag in Flying Birds: Applications to Birdlike Micro Air Vehicles , 2001 .

[22]  Wei Shyy,et al.  Laminar-Turbulent Transition of a Low Reynolds Number Rigid or Flexible Airfoil , 2006 .

[23]  G. Carrier,et al.  On the non-linear vibration problem of the elastic string , 1945 .

[24]  David R. Kincaid,et al.  Numerical mathematics and computing , 1980 .

[25]  K. Isogai,et al.  Effects of Dynamic Stall on Propulsive Efficiency and Thrust of Flapping Airfoil , 1999 .

[26]  Yongsheng Lian,et al.  Three-Dimensional Fluid-Structure Interactions of a Membrane Wing for Micro Air Vehicle Applications , 2003 .

[27]  Dragos Viieru,et al.  Membrane Wing-Based Micro Air Vehicles , 2005 .

[28]  Raymond E. Gordnier,et al.  High fidelity computational simulation of a membrane wing airfoil , 2008 .

[29]  Allen Labryer,et al.  High dimensional harmonic balance dealiasing techniques for a Duffing oscillator , 2009 .

[30]  Wei Shyy,et al.  Membrane wing aerodynamics for micro air vehicles , 2003 .

[31]  E. Dowell,et al.  Chaotic Vibrations: An Introduction for Applied Scientists and Engineers , 1988 .

[32]  Jin-Ho Kim,et al.  Numerical Study on the Unsteady-Force-Generation Mechanism of Insect Flapping Motion , 2008 .

[33]  Sam Heathcote,et al.  Flexible flapping airfoil propulsion at low Reynolds numbers , 2005 .

[34]  Peter J. Attar.  Some results for approximate strain and rotation tensor formulations in geometrically non-linear Reissner Mindlin plate theory , 2008 .