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.

References

References
1.
Hodges
,
D. H.
, and
Pierce
,
G. A.
,
2014
,
Introduction to Structural Dynamics and Aeroelasticity
,
2nd ed.
,
Cambridge University Press
,
Cambridge, UK
.
2.
Lind
,
R.
,
2002
, “
Match-Point Solutions for Robust Flutter Analysis
,”
J. Aircr.
,
39
(
1
), pp.
91
99
.
3.
Borglund
,
D.
,
2004
, “
The μ-k Method for Robust Flutter Solutions
,”
J. Aircr.
,
41
(
5
), pp.
1209
1216
.
4.
Borglund
,
D.
,
2003
, “
Robust Aeroelastic Stability Analysis Considering Frequency-Domain Aerodynamic Uncertainty
,”
J. Aircr.
,
40
(
1
), pp.
189
193
.
5.
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
.
6.
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
.
7.
Irani
,
S.
, and
Sazesh
,
S.
,
2013
, “
A New Flutter Speed Analysis Method Using Stochastic Approach
,”
J. Fluids Struct.
,
40
, pp.
105
114
.
8.
Pons
,
A.
,
2015
, “
Aeroelastic Flutter as a Multiparameter Eigenvalue Problem
,” Master's thesis, University of Canterbury, Christchurch, New Zealand.
9.
Bisplinghoff
,
R. L.
,
Ashley
,
H.
, and
Halfman
,
R. L.
,
1957
,
Aeroelasticity
,
Addison-Wesley
,
Reading, MA
.
10.
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
.
11.
Librescu
,
L.
, and
Song
,
O.
,
2006
,
Thin-Walled Composite Beams
,
Springer
,
Dordrecht, The Netherlands
.
12.
Yates
,
J. E.
,
1970
,
A Study of Panel Flutter With the Exact Method of Zeydel
,
NASA
,
Washington, DC
.
13.
Goland
,
M.
,
1945
, “
The Flutter of a Uniform Cantilever Wing
,”
ASME J. Appl. Mech.
,
12
(
4
), pp.
A197
A208
.
14.
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.
15.
Graham
,
G. M.
, and
Jenkins
,
J. E.
,
1997
, “
Aeroelasticity of an Airfoil Test Rig
,”
J. Fluids Struct.
,
11
(
5
), pp.
485
506
.
16.
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
.
17.
Liska
,
S.
, and
Dowell
,
E. H.
,
2009
, “
Continuum Aeroelastic Model for a Folding-Wing Configuration
,”
AIAA J.
,
47
(
10
), pp.
2350
2358
.
18.
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.
19.
Pons
,
A.
, and
Gutschmidt
,
S.
,
2014
, “
Lower-Wing Flutter in Biplanes
,”
8th European Nonlinear Dynamics Conference
, Vienna, Austria.
20.
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
.
21.
Balakrishnan
,
A. V.
,
2012
,
Aeroelasticity: The Continuum Theory
,
Springer
,
Dordrecht, The Netherlands
.
22.
Karpouzian
,
G.
, and
Librescu
,
L.
,
1996
, “
Nonclassical Effects on Divergence and Flutter of Anisotropic Swept Aircraft Wings
,”
AIAA J.
,
34
(
4
), pp.
786
794
.
23.
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
.
24.
Theodorsen
,
T.
,
1935
, “
General Theory of Aerodynamic Instability and the Mechanism of Flutter
,” Washington, DC, NACA Report No. 496.
25.
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
.
26.
Graham
,
R. L.
,
Knuth
,
D. E.
, and
Patashnik
,
O.
,
1994
,
Concrete Mathematics
,
2nd ed.
,
Addison-Wesley
,
Reading, MA
.
27.
Tang
,
K. T.
,
2006
,
Mathematical Methods for Engineers and Scientists
, Vol.
2
,
Springer
,
Dordrecht, The Netherlands
.
28.
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
.
29.
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
.
30.
Berhanu
,
M.
,
2005
, “
The Polynomial Eigenvalue Problem
,” Doctoral Dissertation, University of Manchester, Manchester, UK.
31.
Magnus
,
J. R.
,
1999
,
Matrix Differential Calculus With Applications in Statistics and Econometrics
,
2nd ed.
,
Wiley
,
Chichester, UK
.
32.
Duffy
,
D. G.
,
2010
,
Advanced Engineering Mathematics With MATLAB
,
3rd ed.
,
CRC Press
,
Boca Raton, FL
.
33.
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
.
34.
Laitone
,
E. V.
,
1952
, “
Theodorsen's Circulation Function for Generalized Motion
,”
Readers' Forum, J. Aeronaut. Sci.
,
19
(
3
), pp.
211
213
.
35.
van de Vooren
,
A. I.
,
1952
, “
Generalization of the Theodorsen Function to Stable Oscillations
,”
Readers' Forum, J. Aeronaut. Sci.
,
19
(
3
), pp.
209
211
.
36.
Mehrmann
,
V.
, and
Voss
,
H.
,
2004
, “
Nonlinear Eigenvalue Problems: A Challenge for Modern Eigenvalue Methods
,”
GAMM
Mitt.
,
27
(
2
), pp.
121
152
.
37.
Voss
,
H.
,
2013
, “
Nonlinear Eigenvalue Problems
,”
Handbook of Linear Algebra
,
L.
Hogben
, ed.,
Chapman & Hall/CRC
,
Boca Raton, FL
.
38.
Wang
,
I.
,
2011
, “
Component Modal Analysis of a Folding Wing
,” Master's thesis, Duke University, Durham, NC.
39.
Borello
,
F.
,
Cestino
,
E.
, and
Frulla
,
G.
,
2010
, “
Structural Uncertainty Effect on Classical Wing Flutter Characteristics
,”
J. Aerosp. Eng.
,
23
(
4
), pp.
327
338
.
40.
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.
41.
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.
42.
Marzocca
,
P.
,
Librescu
,
L.
, and
Silva
,
W. A.
,
2002
, “
Aeroelastic Response and Flutter of Swept Aircraft Wings
,”
AIAA J.
,
40
(
5
), pp.
801
812
.
43.
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
.
44.
Polyanin
,
A. D.
, and
Manzhirov
,
A. V.
,
2007
,
Handbook of Mathematics for Engineers and Scientists
,
Chapman & Hall/CRC
,
Boca Raton, FL
.
45.
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
.
46.
Reissner
,
E.
, and
Stein
,
M.
,
1951
, “
Torsion and Transverse Bending of Cantilever Plates
,” Washington, DC, NACA Technical Note No. 2369.
You do not currently have access to this content.