Abstract

Quasi-periodic (QP) solutions of a damped nonlinear QP Mathieu equation with cubic nonlinearity are investigated by using the incremental harmonic balance (IHB) method with two time scales. The damped nonlinear QP Mathieu equation contains two incommensurate harmonic excitation frequencies, one is a small frequency while the other nearly equals twice the linear natural frequency. It is found that Fourier spectra of QP solutions of the equation consist of uniformly spaced sidebands due to cubic nonlinearity. The IHB method with two time scales, which relates to the two excitation frequencies, is adopted to trace solution curves of the equation in an automatical way and find all frequencies of solutions and their corresponding amplitudes. Effects of parametric excitation are studied in detail. Based on approximation of QP solutions by periodic solutions with a large period, Floquet theory is used to study the stability of QP solutions. Three types of QP solutions can be obtained from the IHB method, which agree very well with results from numerical integration. However, the perturbation method using the double-step method of multiple scales (MMS) obtains only one type of QP solutions since the ratio of the small frequency to the linear natural frequency of the first reduced-modulation equation is nearly 1 in the second perturbation procedure, while the other two types of QP solutions from the IHB method with two time scales do not need the ratio. Furthermore, the results from the double-step MMS are different from those numerical integration and the IHB method with two time scales.

References

1.
Rand
,
R. H.
,
2015
, Lecture Notes on Nonlinear Vibrations (version 53). https://hdl.handle.net/1813/28989
2.
Ruby
,
L.
,
1996
, “
Applications of the Mathieu Equation
,”
Am. J. Phys.
,
64
(
1
), pp.
39
44
.
3.
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
.
4.
Turner
,
K. L.
,
Miller
,
S. A.
,
Hartwell
,
P. G.
,
MacDonald
,
N. C.
,
Strogatz
,
S. H.
, and
Adams
,
S. G.
,
1998
, “
Five Parametric Resonances in a Microelectromechanical System
,”
Nature
,
396
(
6707
), pp.
149
152
.
5.
Raja
,
M. A. Z.
,
Manzar
,
M. A.
,
Shah
,
F. H.
, and
Shah
,
F. H.
,
2018
, “
Intelligent Computing for Mathieu’s Systems for Parameter Excitation, Vertically Driven Pendulum and Dusty Plasma Models
,”
Appl. Soft Comput.
,
62
, pp.
359
372
.
6.
Fernandes
,
J. C.
,
Sebastião
,
P. J.
,
Gonçalves
,
L. N.
, and
Ferraz
,
A.
,
2017
, “
Study of Large-Angle Anharmonic Oscillations of a Physical Pendulum Using an Acceleration Sensor
,”
Eur. J. Phys.
,
38
(
4
), p.
045004
.
7.
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
.
8.
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
.
9.
Kovacic
,
I.
,
Rand
,
R.
, and
Sah
,
S. M.
,
2018
, “
Mathieu’s Equation and Its Generalizations: Overview of Stability Charts and Their Features
,”
Appl. Mech. Rev.
,
70
(
2
), p.
020802
.
10.
Esmailzadeh
,
E.
, and
Nakhaie-Jazar
,
G.
,
1997
, “
Periodic Solution of a Mathieu–Duffing Type Equation
,”
Int. J. Non-Linear Mech.
,
32
(
5
), pp.
905
912
.
11.
Ng
,
L.
, and
Rand
,
R.
,
2002
, “
Bifurcations in a Mathieu Equation With Cubic Nonlinearities
,”
Chaos Solitons Fractals
,
14
(
2
), pp.
173
181
.
12.
Kovacic
,
I.
, and
Cveticanin
,
L.
,
2009
, “
The Effects of Strong Cubic Nonlinearity on the Existence of Periodic Solutions of the Mathieu–Duffing Equation
,”
ASME J. Appl. Mech.
,
76
(
5
), p.
054501
.
13.
Zounes
,
R. S.
, and
Rand
,
R. H.
,
1998
, “
Transition Curves for the Quasi-Periodic Mathieu Equation
,”
SIAM J. Appl. Math.
,
58
(
4
), pp.
1094
1115
.
14.
Rand
,
R.
,
Guennoun
,
K.
, and
Belhaq
,
M.
,
2003
, “
2:2:1 Resonance in the Quasiperiodic Mathieu Equation
,”
Nonlinear Dyn.
,
31
(
4
), pp.
367
374
.
15.
Rand
,
R.
, and
Morrison
,
T.
,
2005
, “
2:1:1 Resonance in the Quasi-Periodic Mathieu Equation
,”
Nonlinear Dyn.
,
40
(
2
), pp.
195
203
.
16.
Sofroniou
,
A.
, and
Bishop
,
S.
,
2014
, “
Dynamics of a Parametrically Excited System With Two Forcing Terms
,”
Mathematics
,
2
(
3
), pp.
172
195
.
17.
Belhaq
,
M.
,
Guennoun
,
K.
, and
Houssni
,
M.
,
2002
, “
Asymptotic Solutions for a Damped Non-Linear Quasi-Periodic Mathieu Equation
,”
Int. J. Non-Linear Mech.
,
37
(
3
), pp.
445
460
.
18.
Guennoun
,
K.
,
Houssni
,
M.
, and
Belhaq
,
M.
,
2002
, “
Quasi-Periodic Solutions and Stability for a Weakly Damped Nonlinear Quasi-Periodic Mathieu Equation
,”
Nonlinear Dyn.
,
27
(
3
), pp.
211
236
.
19.
Zounes
,
R. S.
, and
Rand
,
R. H.
,
2002
, “
Global Behavior of a Nonlinear Quasiperiodic Mathieu Equation
,”
Nonlinear Dyn.
,
27
(
1
), pp.
87
105
.
20.
Abouhazim
,
N.
,
Rand
,
R. H.
, and
Belhaq
,
M.
,
2006
, “
The Damped Nonlinear Quasiperiodic Mathieu Equation Near 2:2:1 Resonance
,”
Nonlinear Dyn.
,
45
(
3–4
), pp.
237
247
.
21.
Davis
,
S. H.
, and
Rosenblat
,
S.
,
1980
, “
A Quasiperiodic Mathieu–Hill Equation
,”
SIAM J. Appl. Math.
,
38
(
1
), pp.
139
155
.
22.
Luongo
,
A.
, and
Zulli
,
D.
,
2011
, “
Parametric, External and Self-Excitation of a Tower Under Turbulent Wind Flow
,”
J. Sound Vib.
,
330
(
13
), pp.
3057
3069
.
23.
Robles
,
L.
,
Ruggero
,
M. A.
, and
Rich
,
N. C.
,
1997
, “
Two-Tone Distortion on the Basilar Membrane of the Chinchilla Cochlea
,”
J. Neurophysiol.
,
77
(
5
), pp.
2385
2399
.
24.
Sharma
,
A.
, and
Sinha
,
S. C.
,
2018
, “
An Approximate Analysis of Quasi-Periodic Systems Via Floquét Theory
,”
J. Comput. Nonlinear Dyn.
,
13
(
2
), p.
021008
.
25.
Huang
,
J. L.
, and
Zhu
,
W. D.
,
2017
, “
An Incremental Harmonic Balance Method With Two Timescales for Guasiperiodic Motion of Nonlinear Systems Whose Spectrum Contains Uniformly Spaced Sideband Frequencies
,”
Nonlinear Dyn.
,
90
(
2
), pp.
1015
1033
.
26.
Huang
,
J. L.
,
Wang
,
T.
, and
Zhu
,
W. D.
,
2021
, “
An Incremental Harmonic Balance Method With Two Time-Scales for Quasi-Periodic Responses of a Van Der Pol–Mathieu Equation
,”
Int. J. Non-Linear Mech.
,
135
, p.
103767
.
27.
Belhaq
,
M.
, and
Houssni
,
M.
,
1999
, “
Quasi-Periodic Oscillations, Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations
,”
Nonlinear Dyn.
,
18
(
1
), pp.
1
24
.
28.
Kreider
,
W.
, and
Nayfeh
,
A. H.
,
1998
, “
Experimental Investigation of Single-Mode Responses in a Fixed-Fixed Buckled Beam
,”
Nonlinear Dyn.
,
15
(
2
), pp.
155
177
.
29.
Huang
,
J. L.
,
Xiao
,
L. J.
, and
Zhu
,
W. D.
,
2020
, “
Investigation of Quasi-Periodic Response of a Buckled Beam Under Harmonic Base Excitation With an “unexplained” Sideband Structure
,”
Nonlinear Dyn.
,
100
(
3
), pp.
2103
2119
.
30.
Liao
,
H.
,
Zhao
,
Q.
, and
Fang
,
D.
,
2020
, “
The Continuation and Stability Analysis Methods for Quasi-Periodic Solutions of Nonlinear Systems
,”
Nonlinear Dyn.
,
100
(
2
), pp.
1469
1496
.
31.
Moore
,
G.
,
2005
, “
Floquet Theory As a Computational Tool
,”
SIAM J. Numer. Anal.
,
42
(
6
), pp.
2522
2568
.
You do not currently have access to this content.