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Research Papers

# Aeroelastic Flutter of Continuous Systems: A Generalized Laplace Transform Method

[+] Author and Article Information
Arion Pons

Department of Engineering,
University of Cambridge,
Trumpington Street,
Cambridge CB2 1PZ, UK

Stefanie Gutschmidt

Department of Mechanical Engineering,
University of Canterbury,
Private Bag 4800,
Christchurch 8140, New Zealand
e-mail: stefanie.gutschmidt@canterbury.ac.nz

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 8, 2016; final manuscript received May 6, 2016; published online May 24, 2016. Assoc. Editor: George Kardomateas.

J. Appl. Mech 83(8), 081005 (May 24, 2016) (8 pages) Paper No: JAM-16-1075; doi: 10.1115/1.4033597 History: Received February 08, 2016; Revised May 06, 2016

## Abstract

This paper presents a generalization of the Laplace transform method (LTM) for determining the flutter points of a linear ordinary-differential aeroelastic system—a linear system involving a spatial derivative as well as a time-eigenvalue parameter. Current implementations of the LTM have two major problems: they are unable to solve systems of arbitrary size, order, and boundary conditions, and they require certain key operations to be performed by hand or with symbolic manipulation libraries. Our generalized method overcomes both these problems. We also devise a new method for solving and visualizing the algebraic system that arises from the LTM procedure. We validate our generalized LTM and novel solution method against both the Goland wing model and a large system of high differential order, as a demonstration of their effectiveness for solving such systems.

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## References

Hodges, D. H. , and Pierce, G. A. , 2014, Introduction to Structural Dynamics and Aeroelasticity, 2nd ed., Cambridge University Press, Cambridge, UK.
Lind, R. , 2002, “ Match-Point Solutions for Robust Flutter Analysis,” J. Aircr., 39(1), pp. 91–99.
Borglund, D. , 2004, “ The μ-k Method for Robust Flutter Solutions,” J. Aircr., 41(5), pp. 1209–1216.
Borglund, D. , 2003, “ Robust Aeroelastic Stability Analysis Considering Frequency-Domain Aerodynamic Uncertainty,” J. Aircr., 40(1), pp. 189–193.
Haddadpour, H. , and Firouz-Abadi, R. D. , 2009, “ True Damping and Frequency Prediction for Aeroelastic Systems: The PP Method,” J. Fluids Struct., 25(7), pp. 1177–1188.
Afolabi, D. , Pidaparti, R. M. V. , and Yang, H. T. Y. , 1998, “ Flutter Prediction Using an Eigenvector Orientation Approach,” AIAA J., 36(1), pp. 69–74.
Irani, S. , and Sazesh, S. , 2013, “ A New Flutter Speed Analysis Method Using Stochastic Approach,” J. Fluids Struct., 40, pp. 105–114.
Pons, A. , 2015, “ Aeroelastic Flutter as a Multiparameter Eigenvalue Problem,” Master's thesis, University of Canterbury, Christchurch, New Zealand.
Bisplinghoff, R. L. , Ashley, H. , and Halfman, R. L. , 1957, Aeroelasticity, Addison-Wesley, Reading, MA.
Dowell, E. H. , Cox, D. , Curtiss, Jr., H. C. , Edwards, J. W. , Hall, K. C. , Peters, D. A. , Scanlan, R. H. , Simiu, E. , Sisto, F. , and Strganac, T. W. , 2004, A Modern Course in Aeroelasticity, Springer Netherlands, Dordrecht, The Netherlands.
Librescu, L. , and Song, O. , 2006, Thin-Walled Composite Beams, Springer, Dordrecht, The Netherlands.
Yates, J. E. , 1970, A Study of Panel Flutter With the Exact Method of Zeydel, NASA, Washington, DC.
Goland, M. , 1945, “ The Flutter of a Uniform Cantilever Wing,” ASME J. Appl. Mech., 12(4), pp. A197–A208.
Runyan, H. L. , and Watkins, C. E. , 1950, “ Flutter of a Uniform Wing With an Arbitrarily Placed Mass According to a Differential-Equation Analysis and a Comparison With Experiment,” Washington, DC, NACA Report No. 966.
Graham, G. M. , and Jenkins, J. E. , 1997, “ Aeroelasticity of an Airfoil Test Rig,” J. Fluids Struct., 11(5), pp. 485–506.
Librescu, L. , and Thangjitham, S. , 1991, “ Analytical Studies on Static Aeroelastic Behavior of Forward-Swept Composite Wing Structures,” J. Aircr., 28(2), pp. 151–157.
Liska, S. , and Dowell, E. H. , 2009, “ Continuum Aeroelastic Model for a Folding-Wing Configuration,” AIAA J., 47(10), pp. 2350–2358.
Pons, A. , and Gutschmidt, S. , 2014, “ Lower-Wing Structural Dynamics and Flutter in V-Strut Biplanes,” 17th U.S. National Congress on Theoretical & Applied Mechanics, East Lansing, MI.
Pons, A. , and Gutschmidt, S. , 2014, “ Lower-Wing Flutter in Biplanes,” 8th European Nonlinear Dynamics Conference, Vienna, Austria.
Chonan, S. , 1986, “ Steady State Response of an Axially Moving Strip Subjected to a Stationary Lateral Load,” J. Sound Vib., 107(1), pp. 155–165.
Balakrishnan, A. V. , 2012, Aeroelasticity: The Continuum Theory, Springer, Dordrecht, The Netherlands.
Karpouzian, G. , and Librescu, L. , 1996, “ Nonclassical Effects on Divergence and Flutter of Anisotropic Swept Aircraft Wings,” AIAA J., 34(4), pp. 786–794.
Qin, Z. , and Librescu, L. , 2003, “ Aeroelastic Instability of Aircraft Wings Modelled as Anisotropic Composite Thin-Walled Beams in Incompressible Flow,” J. Fluids Struct., 18(1), pp. 43–61.
Theodorsen, T. , 1935, “ General Theory of Aerodynamic Instability and the Mechanism of Flutter,” Washington, DC, NACA Report No. 496.
Peters, D. A. , Karunamoorthy, S. , and Cao, W.-M. , 1995, “ Finite State Induced Flow Models. I—Two-Dimensional Thin Airfoil,” J. Aircr., 32(2), pp. 313–322.
Graham, R. L. , Knuth, D. E. , and Patashnik, O. , 1994, Concrete Mathematics, 2nd ed., Addison-Wesley, Reading, MA.
Tang, K. T. , 2006, Mathematical Methods for Engineers and Scientists, Vol. 2, Springer, Dordrecht, The Netherlands.
Aurentz, J. L. , Vandebril, R. , and Watkins, D. S. , 2013, “ Fast Computation of the Zeros of a Polynomial via Factorization of the Companion Matrix,” SIAM J. Sci. Comput., 35(1), pp. A255–A269.
Holzel, M. S. , and Bernstein, D. S. , 2011, “ SVD-Based Computation of Zeros of Polynomial Matrices,” Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, pp. 6962–6966.
Berhanu, M. , 2005, “ The Polynomial Eigenvalue Problem,” Doctoral Dissertation, University of Manchester, Manchester, UK.
Magnus, J. R. , 1999, Matrix Differential Calculus With Applications in Statistics and Econometrics, 2nd ed., Wiley, Chichester, UK.
Duffy, D. G. , 2010, Advanced Engineering Mathematics With MATLAB, 3rd ed., CRC Press, Boca Raton, FL.
Luke, Y. L. , and Dengler, M. A. , 1951, “ Tables of the Theodorsen Circulation Function for Generalized Motion,” Readers' Forum, J. Aeronaut. Sci., 18(7), pp. 478–483.
Laitone, E. V. , 1952, “ Theodorsen's Circulation Function for Generalized Motion,” Readers' Forum, J. Aeronaut. Sci., 19(3), pp. 211–213.
van de Vooren, A. I. , 1952, “ Generalization of the Theodorsen Function to Stable Oscillations,” Readers' Forum, J. Aeronaut. Sci., 19(3), pp. 209–211.
Mehrmann, V. , and Voss, H. , 2004, “ Nonlinear Eigenvalue Problems: A Challenge for Modern Eigenvalue Methods,” GAMM‐Mitt., 27(2), pp. 121–152.
Voss, H. , 2013, “ Nonlinear Eigenvalue Problems,” Handbook of Linear Algebra, L. Hogben , ed., Chapman & Hall/CRC, Boca Raton, FL.
Wang, I. , 2011, “ Component Modal Analysis of a Folding Wing,” Master's thesis, Duke University, Durham, NC.
Borello, F. , Cestino, E. , and Frulla, G. , 2010, “ Structural Uncertainty Effect on Classical Wing Flutter Characteristics,” J. Aerosp. Eng., 23(4), pp. 327–338.
Wang, Y. , Wynn, A. , and Palacios, R. , 2013, “ Robust Aeroelastic Control of Very Flexible Wings Using Intrinsic Models,” 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Boston, MA.
Palacios, R. , and Epureanu, B. , 2011, “ An Intrinsic Description of the Nonlinear Aeroelasticity of Very Flexible Wings,” 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Denver, CO.
Marzocca, P. , Librescu, L. , and Silva, W. A. , 2002, “ Aeroelastic Response and Flutter of Swept Aircraft Wings,” AIAA J., 40(5), pp. 801–812.
Goland, M. , and Luke, Y. L. , 1948, “ The Flutter of a Uniform Wing With Tip Weights,” ASME J. Appl. Mech., 15(1), pp. 13–20.
Polyanin, A. D. , and Manzhirov, A. V. , 2007, Handbook of Mathematics for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, FL.
Jensen, D. W. , and Crawley, E. F. , 1984, “ Frequency Determination Techniques for Cantilevered Plates With Bending-Torsion Coupling,” AIAA J., 22(3), pp. 415–420.
Reissner, E. , and Stein, M. , 1951, “ Torsion and Transverse Bending of Cantilever Plates,” Washington, DC, NACA Technical Note No. 2369.

## Figures

Fig. 1

det(D) against χ for the Goland wing, in vacuo

Fig. 2

Contour plot of the Goland wing system with modal frequency paths superimposed

Fig. 3

Modal damping paths of the Goland wing

Fig. 4

Contour plot of the higher-order test system

Fig. 5

Modal frequency and damping paths of the higher-order test system in the vicinity of the flutter point

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