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Nonlinear Dynamics of Very High Dimensional Fluid-Structural Systems

  • Earl H. Dowell
  • D. M. Tang
Part of the Fluid Mechanics and its Applications book series (FMIA, volume 75)

Abstract

The dynamics of mechanical systems of high dimension or many degrees of freedom will be discussed. The modeling of both fluid and structural systems will be considered with special emphasis given to fluid-structural or aeroelastic systems. Novel and effective theoretical and computational techniques will be considered and illustrative correlations with experimental results given. Also briefly considered will be nanoscale systems that are modeled at the molecular level using similar mathematical techniques. Aeroelastic systems are those that involved the coupled interaction between a convecting fluid flow and a flexible elastic structure. The nonlinear dynamical response of such systems is of great current interest. Currently operational aircraft are known to encounter limit cycle oscillations (LCO) in certain flight regimes and relatively simple experimental wind tunnel models have been designed to exhibit LCO as well. The LCO may be either beneficial or dangerous for the safety of the aircraft. The results of several wind tunnel experiments are discussed and compared to those from mathematical models. The physical models include (1) an airfoil with control surface freeplay; (2) a delta wing with structural geometrical nonlinearities due to plate-like deformations; and (3) a very high aspect ratio wing with geometrical structural nonlinearities due to coupling among torsional twist, transverse bending and fore-and-aft bending. In addition, the theoretical and computational advantages of modeling the aerodynamic flow field in terms of a set of global modes and also using a novel form of the harmonic balance method are emphasized. A recent theoretical result for large shock motions in a viscous transonic flow around an oscillating airfoil undergoing LCO is presented to illustrate the results to be obtained by such methods.

Keywords

Mach Number AIAA Paper Harmonic Balance Method Limit Cycle Oscillation Flutter Speed 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Cunningham, A.M., Jr., “A Generic Nonlinear Aeroelastic Method with Semi-Empirical Nonlinear Unsteady Aerodynamics,” Vol. 1 and 2, AFRL-VA-WP-TR-1999–3014, 1999.Google Scholar
  2. 2.
    Friedmann, P.P.,“Renaissance of Aeroelasticity and Its Future,” Journal of Aircraft, Vol.36, No.1, pp. 105–121, 1999.Google Scholar
  3. 3.
    Woolston, D.S., Runyan, H.L. and Andrews, R.E., “An Investigation of Effects of Certain Types of Structural Nonlinearities on Wing and Control Surface Flutter,” Journal of Aeronautical Sciences, Vol.24, 1957, pp. 57–63.Google Scholar
  4. 4.
    Shen, S.F., “An Approximate Analysis of Nonlinear Flutter Problems,” Journal of Aeronautical Sciences, Vol.26, 1959, pp. 25.MATHGoogle Scholar
  5. 5.
    Breitbach, E., “Effects of Structural Nonlinearities on Aircraft Vibration and Flutter,” AGARD Technical Report 665, 1977.Google Scholar
  6. 6.
    Tang, D.M., and Dowell, E.H., “Comparison of Theory and Experiment for Nonlinear Flutter and Stall Response of a Helicopter Blade,” Journal of Sound and Vibration, Vol. 165(2), 1993, pp. 251–276.CrossRefGoogle Scholar
  7. 7.
    Brase, L.O. and Eversman, W., “Application of Transient Aerodynamics to the Structural Nonlinear Flutter Problem,” Journal of Aircraft, Vol. 25, No. 11, 1988, pp. 1060–1068.CrossRefGoogle Scholar
  8. 8.
    Yang, Z.C. and Zhao, L.C., “Analysis of Limit Cycle Flutter of an Airfoil in incompressible Flow,” Journal of Sound and Vibration, Vol 123, 1988, pp. 1–13.CrossRefGoogle Scholar
  9. 9.
    Yang, Z.C. and Zhao, L.C., “Chaotic Motions of an Airfoil with Nonlinear Stiffness in Incompressible Flow,” Journal of Sound and Vibration, Vol. 138, 1990, pp. 245–254.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hauenstein, A.J., Zara, J.A., Eversman, W. and Qumei, I., “Chaotic and Nonlinear Dynamic Response of Aerosurfaces with Structural Nonlinearities,” AIAA-92-2547-CP, 1992.Google Scholar
  11. 11.
    Liu, J.K. and Zhao, L.C., “Bifurcation Analysis of Airfoil in Incompressible Flow,” Journal of Sound and Vibration, Vol. 154, 1992, pp. 117–124.MATHCrossRefGoogle Scholar
  12. 12.
    Tang, D.M., and Dowell, E.H., “Chaotic Stall Response of a Helicopter in Forward Flight,” Journal of Fluids and Structures, Vol. 6, No. 3, 1992, pp. 311–335.CrossRefGoogle Scholar
  13. 13.
    Tang, D.M., and Dowell, E.H., “Flutter and Stall Response of Helicopter Blade with Structural Nonlinearity,” Journal of Aircraft, Vol.29, No. 5 Sept-Oct, 1992, pp. 953–960.CrossRefGoogle Scholar
  14. 14.
    Tang, D.M. and Dowell, E.H., “Nonlinear Aeroelasticity in Rotorcraft,” Journal of Mathematical and Computer Modeling, Vol. 18, No. 3/4, 1993, pp. 157–184.Google Scholar
  15. 15.
    Tang, D.M., and Dowell, E.H.,“Experimental and Theoretical Study for Nonlinear Aeroelastic Behavior of a Flexible Rotor Blade,” AIAA Journal, Vol. 31(6) June 1993, pp. 1133–1142.CrossRefGoogle Scholar
  16. 16.
    Price, S.J., Alighanbari, H. and Lee, B.H.K., “Post-Instability Behavior of a Two-Dimensional Airfoil with a Structural Nonlinearities,” Journal of Aircraft, Vol. 31, 1994, pp. 1395.CrossRefGoogle Scholar
  17. 17.
    Price, S.J., Alighanbari, H. and Lee, B.H.K., “The Aeroelastic Response of a Two-Dimensional Airfoil with Bilinear and Cubic Structural Nonlinearities,” Journal of Fluids and Structures, Vol. 9, 1995, pp. 175–193.CrossRefGoogle Scholar
  18. 18.
    Tang, D.M., and Dowell, E.H., “Response of a Non-rotating Rotor Blade to Lateral Turbulence in Sinusoidal Pulsating Flow, Part 2: Experiment,” Journal of Aircraft, Vol. 32, No.1, Jan-Feb, 1995, pp. 154–160.CrossRefGoogle Scholar
  19. 19.
    Conner, M.D., Virgin, L.N., and Dowell, E.H., “A Note on Accurate Numerical Integration of State-Space Models for Aeroelastic System with Freeplay,” AIAA Journal, Vol. 34, 1996, pp.2202.CrossRefGoogle Scholar
  20. 20.
    Kim, S.H. and Lee, I., “Aeroelastic Analysis of a Flexible Airfoil with a Freeplay Nonlinearity,” Journal of Sound and Vibration, Vol. 195, 1996, pp. 823–846.Google Scholar
  21. 21.
    Lee, B.H.K. and Leblanc, P., “ Flutter Analysis of a Two-Dimensional Airfoil with Cubic Nonlinear Restoring Force,” National Research Council of Canada, Aeronautical Note NAE-AN-36, NRC No. 25438, 1996.Google Scholar
  22. 22.
    O’Neil, T., Gilliat, H. and Strganac, T., “Investigation of Aeroelastic Response for a System with Continuous Structural Nonlinearities,” AIAA Paper 96–1390, 1996.Google Scholar
  23. 23.
    Tang, D.M., and Dowell, E.H., “Nonlinear Response of a Non-Rotating Rotor Blade to a Periodic Gust,” Journal of Fluids and Structures, Vol. 10, No. 7, 1996, pp. 721–742.CrossRefGoogle Scholar
  24. 24.
    Conner, M.C., Tang, D.M., Dowell, E.H. and Virgin, L.N., “Nonlinear Behavior of A Typical Airfoil Section with Control Surface Freeplay: A Numerical and Experimental Study,” Journal of Fluids and Structures, 11, 1997, pp. 89–112.CrossRefGoogle Scholar
  25. 25.
    Lee, B.H.K., Gong L. and Wong Y.S., “Analysis and Computation of Nonlinear Dynamic Response of a Two-Degree-of-Freedom System and Its Application in Aeroelasticity,” Journal of Fluids and Structures, Vol. 11, 1997, pp. 225–2467.CrossRefGoogle Scholar
  26. 26.
    Tang, D.M., Dowell, E.H. and Virgin, L.N., “Limit Cycle Behavior of an Airfoil with a Control Surface,” Journal of Fluids and Structures, Vol. 12, No. 7, 1998, pp. 839–858.CrossRefGoogle Scholar
  27. 27.
    Lee, B.H.K., Jiang, L.Y. and Wong Y.S., “Flutter of An Airfoil with a Cubic Nonlinear Restoring Force,” Journal of Fluids and Structures, Vol. 13, 1999, pp. 75–101.CrossRefGoogle Scholar
  28. 28.
    Lee, B.H.K., Price, S.J. and Wong Y.S., “Nonlinear Aeroelastic Analysis of Airfoils: Bifurcation and Chaos,” Progress in Aerospace Sciences, Vol. 35, No. 3, 1999, pp. 205–334.CrossRefGoogle Scholar
  29. 29.
    Kim, D.H. and Lee, I., “Transonic and Low-Supersonic Aeroelastic Analysis of a Two-Degree-of-Freedom Airfoil with a Freeplay Nonlinearity,” Journal of Sound and Vibration, Vol. 234, No. 5, 2000, pp. 859–880.CrossRefGoogle Scholar
  30. 30.
    Liu, L., Wong, Y.S. and Lee, B.H.K., “Application of the Center Manifold Theory in Nonlinear Aeroelasticity,” Journal of Sound and Vibration, Vol. 234, No. 4, 2000, pp. 641–659.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Dowell, E.H., Thomas, J.P. and Hall, K.C., “Transonic Limit Cycle Oscillation Analysis Using Reduced Order Modal Aerodynamic Models,” AIAA Paper 2001-1212, presented at the 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Seattle, Washington, April 16–19, 2001.Google Scholar
  32. 32.
    Von Karman, T. and Sears, W.R., “Airfoil Theory for Non-Uniform Motion,” Journal of Aeronautical Sciences, Vol. 5, No. 10, August 1938, pp. 379–390.MATHGoogle Scholar
  33. 33.
    Dowell, E.H., “Aeroelasticity of Plates and Shells,” Kluwer Academic Publishers, 1975.Google Scholar
  34. 34.
    Mei, C., Abdel-Motagaly, K. and Chen, R.R., “Review of Nonlinear Panel Flutter at Supersonic and Hypersonic Speeds,” Applied Mechanics Reviews, Vol. 52, No. 10, October 1999, pp. 321–332.CrossRefGoogle Scholar
  35. 35.
    Dowell, E.H. and Hall, K.C., “Modeling of Fluid-Structure Interaction,” Annual Review of Fluid Mechanics, Vol. 33, 2001, pp. 445–490.CrossRefGoogle Scholar
  36. 36.
    Hall, K.C., “Eigenanalysis of Unsteady Flows About Airfoils, Cascades, and Wings,” AIAA Journal, Vol. 32, No. 12, 1994, pp. 2426–2432.MATHCrossRefGoogle Scholar
  37. 37.
    Dowell, E.H., “Eigenmode Analysis in Unsteady Aerodynamics: Reduced Order Models,” AIAA Journal, Vol. 34, No. 8, 1996, pp. 1578–1588.MATHCrossRefGoogle Scholar
  38. 38.
    Tang, D.M., Dowell, E.H. and Hall, K.C., “Limit Cycle Oscillations of A Cantilevered Wing in Low Subsonic Flow,” AIAA Journal, Vol. 37, No. 3, 1999, pp. 364–371.CrossRefGoogle Scholar
  39. 39.
    Tang, D.M., Herry, J.K. and Dowell, E.H., “Limit Cycle Oscillations of Delta Wing Models in Low Subsonic Flow,” AIAA Journal, Vol. 37, No. 11, 1999, pp. 1355–1362.CrossRefGoogle Scholar
  40. 40.
    Tang, D.M., Herry, J.K. and Dowell, E.H., “Response of A Delta Wing Model to A Periodic Gust in Low Subsonic Flow,” Journal of Aircraft, Vol. 37, No. 1, 2000, pp. 155–164.CrossRefGoogle Scholar
  41. 41.
    Dowell, E.H. and Tang, D., “Nonlinear Aeroelasticity and Unsteady Aerodynamics,” AIAA Journal, Vol. 40, No. 9, 2002, pp. 1697–1707.CrossRefGoogle Scholar
  42. 42.
    Gordnier, R.E. and Melville, R.B.,“Physical Mechanisms for Limit-Cycle Oscillations of a Cropped Delta Wing,” AIAA Paper 99–3796, Norfolk, VA, June 1999.Google Scholar
  43. 43.
    Gordnier, R.E. and Melville, R.B., “Numerical Simulation of Limit-Cycle Oscillations of a Cropped Delta Wing Using the Full Navier-Stokes Equations,” International Journal of Computational Fluid Dynamics, 14(3):211–224,2001.MATHCrossRefGoogle Scholar
  44. 44.
    Schairer, E.T. and Hand, L.A., “Measurement of Unsteady Aeroelastic Model Deformation by Stereo Photogrammetry,” AIAA Paper 97–2217, June 1997.Google Scholar
  45. 45.
    Preidikman, S. and Mook, D.T., “Time Domain Simulations of Linear and Nonlinear Aeroelastic Behavior,” Journal of Vibration and Control, Vol. 6, No. 8, pp. 1135–1175,2000.CrossRefGoogle Scholar
  46. 46.
    Von Karman, T., “Encyklopadie der Mathematischen Wissenschaften,” Vol.IV, p. 349, 1910.Google Scholar
  47. 47.
    Hodges, D.H., and Dowell, E.H, “Nonlinear Equations of Motion for the Elastic Bending and Torsion of Twisted Non-Uniform Rotor Blades,” NASA TN D-7818, 1974.Google Scholar
  48. 48.
    Tran, C.T. and Petot, D., “Semi-Empirical Model for the Dynamic Stall of Airfoils in View to the Application to the Calculation of Responses of a Helicopter Blade in Forward Flight,” Vertica, Vol. 5, No. 2, 1981, pp. 35–53.Google Scholar
  49. 49.
    Patil, M.J., Hodges, D.H. and Cesnik, C.E.S, “Limit Cycle Oscillations in High-Aspect-Ratio Wings,” Journal of Fluids and Structures, Vol. 15, No. 1,2001, pp. 107–132.CrossRefGoogle Scholar
  50. 50.
    Patil, M.J., Hodges, D.H. and Cesnik, C.E.S, “Nonlinear Aeroelastic Analysis of Complete Aircraft in Subsonic Flow,” Journal of Aircraft, Vol. 37, No. 5, Sept-Oct 2000, pp. 753–760.CrossRefGoogle Scholar
  51. 51.
    Patil, M.J. and Hodges, D.H., “On the Importance of Aerodynamic and Structural Geometrical Nonlinearities on Aeroelastic Behavior of High-Aspect-Ratio Wings,” AIAA Paper-2000–1448, 2000.Google Scholar
  52. 52.
    Tang, D.M. and Dowell, E.H., “Experimental and Theoretical Study on Flutter and Limit Cycle Oscillations of High-Aspect Ratio Wings,” AIAA Journal, Vol. 39, No. 8, August 2001, pp. 1430–1441.Google Scholar
  53. 53.
    Kim, K. and Strganac, T., “Aeroelastic Studies of a Cantilever Wing with Structural and Aerodynamic Nonlinearities,” AIAA Paper 2002-1412, 43rd AIAA/ASME/ASCE/AHS/ACS Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002.Google Scholar
  54. 54.
    Crespo da Silva, M.R.M. and Glynn, C.C., “Nonlinear Flexural-Torsional Dynamics of Inextensional Beams-I: Equations of Motions,” Journal of Structural Mechanics, Vol. 6, No. 4, pp. 437–448, 1978.CrossRefGoogle Scholar
  55. 55.
    Yurkovich, R.N., Liu, D.D. and Chen, P.C., “The State-of-the-Art of Unsteady Aerodynamics for High Performance Aircraft,” AIAA Paper 2001-0428, January 2001.Google Scholar
  56. 56.
    Dowell, E.H. and Hall, K.C., “Modeling of Fluid-Structure Interaction,” Annual Review of Fluid Mechanics, 33:445–90, 2001.CrossRefGoogle Scholar
  57. 57.
    Bennett, R.M. and Edwards, J.W., “An Overview of Recent Developments in Computational Aeroelasticity,” AIAA Paper No. 98–2421, presented at the AIAA Fluid Dynamics Conference, Albuquerque, NM, June 1998.Google Scholar
  58. 58.
    Beran, P. and Silva, W.,“Reduced-Order Modeling: New Approaches for Computational Physics,” AIAA Paper 2001-0853, 39th Aerospace Sciences Meeting and Exhibit, Reno, NV, January 2001.Google Scholar
  59. 59.
    Kim, T. and Bussoletti, J.E., “An Optimal Reduced Order Aeroelastic Modeling Based on a Response-Based Modal Analysis of Unsteady CFD Models,” AIAA Paper 2001-1525, 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference & Exhibit, Seattle, WA, April 2001.Google Scholar
  60. 60.
    Silva, W.A., “Application of Nonlinear Systems Theory to Transonic Unsteady Aerodynamic Responses,” Journal of Aircraft, Vol. 30, No. 5, pp. 660–668, 1993.CrossRefGoogle Scholar
  61. 61.
    Silva, W.A., “Extension of a Nonlinear System Theory to General-Frequency Unsteady Transonic Aerodynamic Responses,” 34th AIAA Structures, Structural Dynamics, and Materials Conference, Reston, VA, pp. 2490–2503, 1993.Google Scholar
  62. 62.
    Silva, W.A., “Extension of a Nonlinear Systems Theory to Transonic Unsteady Aerodynamic Responses,” AIAA Paper 93–1590, April 1993.Google Scholar
  63. 63.
    Silva, W.A., #x201C;Discrete-Time Linear and Nonlinear Aerodynamic Impulse Responses for Efficient (CFD) Analyses,” PhD Thesis, College of William Mary, Williamsburg, VA, October 1997.Google Scholar
  64. 64.
    Silva, W.A., “Identification of Linear and Nonlinear Aerodynamic Impulse Response Using Digital Filter Techniques,” AIAA Atmospheric Flight Mechanics Conference, Reston, VA, pp. 584–597, 1997.Google Scholar
  65. 65.
    Silva, W.A.,“Reduced-Order Models Based on Linear and Nonlinear Aerodynamic Impulse Response,” International Forum on Aeroelasticity and Structural Dynamics, NASA Langley Research Center, Hampton, VA, pp. 369–379, June 1999.Google Scholar
  66. 66.
    Raven, D., Levy, Y. and Karpel, M., “Aircraft Aeroelastic Analysis and Design Using CFD-Based Unsteady Loads,” AIAA Paper 2000–1325, 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, Atlanta, GA, April 2000.Google Scholar
  67. 67.
    Raveh, D.E., “Reduced-Order Models for Nonlinear Unsteady Aerodynamics,” AIAA Journal, Vol. 39, No. 8, pp. 1417–1429, August 2001.CrossRefGoogle Scholar
  68. 68.
    Farhat, C., Geuzaine, P., Brown, G. and Harris, C., “Nonlinear Flutter Analysis of an F-16 in Stabilized, Accelerated, and Increased Angle of Attack Flight Conditions,” AIAA Paper 2002–1490, April 2002.Google Scholar
  69. 69.
    Farhat, C., Harris, C. and Rixen, D., “Expanding a Flutter Envelope Using Accelerated Flight Data: Application to An F-16 Fighter Configuration,” AIAA Paper 2000–1702, April 2000.Google Scholar
  70. 70.
    Kholodar, D.B., Thomas, J.P., Dowell, E.H. and Hall, K.C., “A Parametric Study of Transonic Airfoil Flutter and Limit Cycle Oscillation Behavior,” AIAA Paper 2002–1211, presented at the AIAA/ASME/ASCE/AHS SDM Conference, Denver, CO, April 2002.Google Scholar
  71. 71.
    Knipfer, A. and Schewe, G., “Investigations of and Oscillation Supercritical 2-D Wing Section in a Transonic Flow,” AIAA Paper No. 99-0653, 36th Aerospace Sciences Meeting and Exhibit, January 1999.Google Scholar
  72. 72.
    Schewe, G. and Deyhle, H., “Experiments on Transonic Flutter of a Two-Dimensional Supercritical Wing with Emphasis on the Nonlinear Effects,” Proceedings of the Royal Aeronautical Society Conference on Unsteady Aerodynamics, London, U.K., July 17-18, 1996.Google Scholar
  73. 73.
    Schewe, G., Knipfer, A. and Henke, H., “Experimentelle und numerisch Untersuchung zum transonischen Flügelflattern im Hinblick auf nichtlineare Effecte,” unpublished manuscript, February 1999.Google Scholar
  74. 74.
    Schewe, G., Knipfer, A., Mai, H. and Dietz, G., “Experimental and Numerical Investigation of Nonlinear Effects in Transonic Flutter,” English Version (Translated by Dr. W.F. King III), German Aerospace Center DLR Final Report Number DLR IB 232-2002 J 01, Corresponds to Final Report for BMBF: Nichtlineare Effekte beim transsonischen Flattern (FKZ 13 N 7172), and Internal Report DLR IB 2001 J03, January 25, 2002.Google Scholar
  75. 75.
    Thomas, J.P., Dowell, E.H. and Hall, K.C., “Modeling Viscous Transonic Limit Cycle Oscillation Behavior Using a Harmonic Balance Approach,” AIAA Paper 2002–1414, presented at 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, Denver, CO, April 22-25, 2002.Google Scholar
  76. 76.
    Weber, S., Jones, K.D., Ekaterinaris, J.A. and Platzer, M.F.,“Transonic Flutter Computations for a 2-D Supercritical Wing,” AIAA Paper 99-0798, 36th Aerospace Sciences Meeting and Exhibit, Reno, NV, January 1999.Google Scholar
  77. 77.
    Tang, L., Bartels, R.E., Chen, P.C. and Liu, D.D., “Simulation of Transonic Limit Cycle Oscillations Using a CFD Time-Marching Method,” AIAA Paper 2001–1290, 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Seattle. WA, April 2001.Google Scholar
  78. 78.
    Castro, B. M., Ekaterinaris, J. A., and Platzer, M. F., “Navier-Stokes Analysis of Wind-Tunnel Interference on Transonic Airfoil Flutter,” AIAA Journal, Vol. 40, No. 7, pp. 1269–1276, July 2002.CrossRefGoogle Scholar
  79. 79.
    Thomas, J.P., Dowell, E.H. and Hall, K.C., “Modeling Limit Cycle Oscillations for an NLR 7301 Airfoil Aeroelastic Configuration Including Correlation with Experiment,” AIAA Paper 2003–1429, presented at 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, April 7-10, 2003.Google Scholar
  80. 80.
    Thomas, J.P., Dowell, E.H. and Hall, K.C., “A Harmonic Balance Approach for Modeling Three-Dimensional Nonlinear Unsteady Aerodynamics and Aeroelasticity,” IMECE-2002-32532, Presented at the ASME International Mechanical Engineering Conference and Exposition, November 2002, New Orleans, Louisiana.Google Scholar
  81. 81.
    Edwards, J.W., “Calculated Viscous and Scale Effects on Transonic Aeroelasticity,” AGARD-R-822, Numerical Unsteady Aerodynamic and Aeroelastic Simulation, pp. 1–1-1-1, March 1998.Google Scholar
  82. 82.
    Thomas, J.P. all, K.C. and Dowell, E.H., “A Harmonic Balance Approach for Modeling Nonlinear Aeroelastic Behavior of Wings in Transonic Viscous Flow,” AIAA Paper 2003-1984, Presented at 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, April 7–10, 2003.Google Scholar
  83. 83.
    Rivera, J.A., et al.,“NACA 0012 Benchmark Model Experimental Flutter Results with Unsteady Pressure Distributions,” NASA TM 107581, March 1992.Google Scholar
  84. 84.
    Kholodar, D.B., Dowell, E.H., Thomas, J.P. and Hall, K.C., “Improved Understanding of Transonic Flutter: A Three Parameter Flutter Surface,” Submitted for publication to the Journal of Aircraft, 2003.Google Scholar
  85. 85.
    Bendiksen, O.O.,“Improved Similarity Rules for Transonic Flutter,” AIAA Paper 99–1350, 40th AIAA/ASME/ASCE/AHS SDM Conference, St. Louis, MO, April 1999.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • Earl H. Dowell
    • 1
  • D. M. Tang
    • 1
  1. 1.Duke UniversityUSA

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