Solutions to the linear unforced Mathieu equation, and their stabilities, are investigated. Floquet theory shows that the solution can be written as a product between an exponential part and a periodic part at the same frequency or half the frequency of excitation. In the current work, an approach combining Floquet theory with the harmonic balance method is investigated. A Floquet solution having an exponential part with an unknown exponential argument and a periodic part consisting of a series of harmonics is assumed. Then, performing harmonic balance, frequencies of the response are found and stability of the solution is examined over a parameter set. The truncated solution is consistent with an existing infinite series solution for the undamped case. The truncated solution is then applied to the damped Mathieu equation and parametric excitation with two harmonics.

References

References
1.
Ruby
,
L.
,
1996
, “
Applications of the Mathieu Equation
,”
Am. J. Phys.
,
64
(
1
), pp.
39
44
.
2.
Li
,
Y.
,
Fan
,
S.
,
Guo
,
Z.
,
Li
,
J.
,
Cao
,
L.
, and
Zhuang
,
H.
,
2013
, “
Mathieu Equation With Application to Analysis of Dynamic Characteristics of Resonant Inertial Sensors
,”
Commun. Nonlinear Sci. Numer. Simul.
,
18
(
2
), pp.
401
410
.
3.
Sofroniou
,
A.
, and
Bishop
,
S.
,
2014
, “
Dynamics of a Parametrically Excited System With Two Forcing Terms
,”
Mathematics
,
2
(
3
), pp.
172
195
.
4.
Ramakrishnan
,
V.
, and
Feeny
,
B. F.
,
2012
, “
Resonances of a Forced Mathieu Equation With Reference to Wind Turbine Blades
,”
ASME J. Vib. Acoust.
,
134
(
6
), p.
064501
.
5.
Inoue
,
T.
,
Ishida
,
Y.
, and
Kiyohara
,
T.
,
2012
, “
Nonlinear Vibration Analysis of the Wind Turbine Blade (Occurrence of the Superharmonic Resonance in the Out of Plane Vibration of the Elastic Blade)
,”
ASME J. Vib. Acoust.
,
134
(
3
), p.
031009
.
6.
Rhoads
,
J. F.
,
Miller
,
N. J.
,
Shaw
,
S. W.
, and
Feeny
,
B. F.
,
2008
, “
Mechanical Domain Parametric Amplification
,”
ASME J. Vib. Acoust.
,
130
(
6
), p.
061006
.
7.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
2008
,
Nonlinear Oscillations
,
Wiley
,
New York
.
8.
Taylor
,
J. H.
, and
Narendra
,
K. S.
,
1969
, “
Stability Regions for the Damped Mathieu Equation
,”
SIAM J. Appl. Math.
,
17
(
2
), pp.
343
352
.
9.
Younesian
,
D.
,
Esmailzadeh
,
E.
, and
Sedaghati
,
R.
,
2005
, “
Existence of Periodic Solutions for the Generalized Form of Mathieu Equation
,”
Nonlinear Dyn.
,
39
(
4
), pp.
335
348
.
10.
Thomson
,
W.
,
1996
,
Theory of Vibration With Applications
,
CRC Press
,
Englewood Cliffs, NJ
.
11.
Nayfeh
,
A. H.
,
2008
,
Perturbation Methods
,
Wiley
,
New York
.
12.
Benaroya
,
H.
, and
Nagurka
,
M. L.
,
2011
,
Mechanical Vibration: Analysis, Uncertainties, and Control
,
CRC Press
,
Englewood Cliffs, NJ
.
13.
Turrittin
,
H.
,
1952
, “
Asymptotic Expansions of Solutions of Systems of Ordinary Linear Differential Equations Containing a Parameter
,”
Contributions to the Theory of Nonlinear Oscillations
, Vol.
II
,
Princeton University Press
,
Princeton, NJ
, pp.
81
116
.
14.
Rand
,
R. H.
,
1969
, “
On the Stability of Hill's Equation With Four Independent Parameters
,”
ASME J. Appl. Mech.
,
36
(
4
), pp.
885
886
.
15.
Ishida
,
Y.
,
Inoue
,
T.
, and
Nakamura
,
K.
,
2009
, “
Vibration of a Wind Turbine Blade (Theoretical Analysis and Experiment Using a Single Rigid Blade Model)
,”
J. Environ. Eng.
,
4
(
2
), pp.
443
454
.
16.
Ecker
,
H.
,
2009
, “
Parametric Excitation in Engineering Systems
,”
20th International Congress of Mechanical Engineering
(
COBEM 2009
), Gramado, Brazil, Nov. 15–20, pp.
15
20
.
17.
Ecker
,
H.
,
2011
, “
Beneficial Effects of Parametric Excitation in Rotor Systems
,”
IUTAM Symposium on Emerging Trends in Rotor Dynamics
, New Delhi, India, Mar. 23–26, pp.
361
371
.
18.
Klotter
,
K.
, and
Kotowski
,
G.
,
1943
, “
Über die Stabilität der Lösungen Hillscher Differentialgleichungen mit drei unabhängigen Parametern
,”
ZAMM
,
23
(
3
), pp.
149
155
.
19.
Stoker
,
J. J.
,
1950
,
Nonlinear Vibrations in Mechanical and Electrical Systems
, Vol.
2
,
Interscience Publishers
,
New York
.
20.
Ward
,
M.
,
2010
, “
Lecture Notes on Basic Floquet Theory
,” http://www.emba.uvm.edu/~jxyang/teaching/
21.
McLachlan
,
N. W.
,
1961
,
Theory and Application of Mathieu Functions
,
Dover
,
New York
.
22.
Peterson
,
A.
, and
Bibby
,
M.
,
2013
,
Accurate Computation of Mathieu Functions
,
Morgan & Claypool Publishers
,
San Rafael, CA
.
23.
Hodge
,
D.
,
1972
,
The Calculation of the Eigenvalues and Eigenfunctions of Mathieu's Equation
, Vol.
1937
,
National Aeronautics and Space Administration
, Washington, DC.
24.
Hagedorn
,
P.
,
1988
,
Non-Linear Oscillations
,
Clarendon Press
,
Oxford, UK
.
25.
Hale
,
J. K.
,
1963
,
Oscillations in Nonlinear Systems
,
McGraw-Hill
,
New York
.
26.
Hayashi
,
C.
,
2014
,
Nonlinear Oscillations in Physical Systems
,
Princeton University Press
,
Princeton, NJ
.
27.
Schmidt
,
G.
, and
Tondl
,
A.
,
1986
,
Non-Linear Vibrations
, Vol.
66
,
Cambridge University Press
,
Cambridge, UK
.
28.
Hartog
,
J. P. D.
,
1985
,
Mechanical Vibrations
,
Dover
,
New York
.
29.
Magnus
,
W.
, and
Winkler
,
S.
,
1979
,
Hill's Equation
,
Dover
,
New York
.
30.
Rand
,
R.
,
2012
, “
Lecture Notes on Nonlinear Vibrations
,” http://www.tam.cornell.edu/randdocs
31.
Whittaker
,
E. T.
,
1913
, “
On the General Solution of Mathieu's Equation
,”
Proc. Edinburgh Math. Soc.
,
32
, pp.
75
80
.
32.
Malasoma
,
J.-M.
,
Lamarque
,
C.-H.
, and
Jezequel
,
L.
,
1994
, “
Chaotic Behavior of a Parametrically Excited Nonlinear Mechanical System
,”
Nonlinear Dyn.
,
5
(
2
), pp.
153
160
.
33.
Coisson
,
R.
,
Vernizzi
,
G.
, and
Yang
,
X.
,
2009
, “
Mathieu Functions and Numerical Solutions of the Mathieu Equation
,”
IEEE International Workshop on Open-Source Software for Scientific Computation
(
OSSC
), Guiyang, China, Sept. 18–20, pp.
3
10
.
34.
Insperger
,
T.
, and
Stépán
,
G.
,
2002
, “
Stability Chart for the Delayed Mathieu Equation
,”
Proc. R. Soc. London, Ser. A,
458
(
2024
), pp.
1989
1998
.
35.
Insperger
,
T.
, and
Stépán
,
G.
,
2003
, “
Stability of the Damped Mathieu Equation With Time Delay
,”
ASME J. Dyn. Syst., Meas., Control
,
125
(
2
), pp.
166
171
.
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