A simple nonlinear undamped Duffing oscillator has been studied for its snap-through behavior at large-amplitude vibrations. We present an algorithm that uses the harmonic balance (HB) method to find amplitude and frequency relationships in two- and three-term approximations for solutions that lie outside the separatrix in the phase space. Trends of the approximate solution properties are examined with reference to an analysis of the limit as the trajectory approaches the separatrix.

References

1.
Harne
,
R. L.
, and
Wang
,
K. W.
,
2014
, “
On the Fundamental and Superharmonic Effects in Bistable Energy Harvesting
,”
J. Intell. Mater. Syst. Struct.
,
25
(
8
), pp.
937
950
.10.1177/1045389X13502856
2.
Harne
,
R. L.
, and
Wang
,
K. W.
,
2013
, “
A Review of the Recent Research on Vibration Energy Harvesting Via Bistable Systems
,”
Smart Mater. Struct.
,
22
(
2
), p.
023001
.10.1088/0964-1726/22/2/023001
3.
Stanton
,
S. C.
,
McGehee
,
C. C.
, and
Mann
,
B. P.
,
2010
, “
Nonlinear Dynamics for Broadband Energy Harvesting: Investigation of a Bistable Piezoelectric Inertial Generator
,”
Phys. D
,
239
(
10
), pp.
640
653
.10.1016/j.physd.2010.01.019
4.
Erturk
,
A.
, and
Inman
,
D. J.
,
2011
, “
Broadband Piezoelectric Power Generation on High-Energy Orbits of the Bistable Duffing Oscillator With Electromechanical Coupling
,”
J. Sound Vib.
,
330
(10), pp.
2339
2353
.10.1016/j.jsv.2010.11.018
5.
Masana
,
R.
, and
Daqaq
,
M. F.
,
2012
, “
Energy Harvesting in the Super-Harmonic Frequency Region of a Twin-Well Oscillator
,”
J. Appl. Phys.
,
111
(
4
), p.
044501
.10.1063/1.3684579
6.
Panyam
,
M.
,
Masana
,
R.
, and
Daqaq
,
M. F.
,
2014
, “
On Approximating the Effective Bandwidth of Bi-Stable Energy Harvesters
,”
Int. J. Non-Linear Mech.
,
67
, pp.
153
163
.10.1016/j.ijnonlinmec.2014.09.002
7.
Manevitch
,
L. I.
,
Sigalov
,
G.
,
Romeo
,
F.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2014
, “
Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Analytical Study
,”
ASME J. Appl. Mech.
,
81
(
4
), p.
041011
.10.1115/1.4025150
8.
Romeo
,
F.
,
Sigalov
,
G.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2015
, “
Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Numerical Study
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
1
), p.
011007
.10.1115/1.4027224
9.
AL-Shudeifat
,
M. A.
,
2014
, “
Highly Efficient Nonlinear Energy Sink
,”
Nonlinear Dynam.
,
76
(
4
), pp.
1905
1920
.10.1007/s11071-014-1256-x
10.
Carrella
,
A.
,
Brennan
,
M. J.
,
Waters
,
T. P.
, and
Shin
,
K.
,
2008
, “
On the Design of a High-Static–Low-Dynamic Stiffness Isolator Using Linear Mechanical Springs and Magnets
,”
J. Sound Vib.
,
315
(
3
), pp.
712
720
.10.1016/j.jsv.2008.01.046
11.
Liu
,
X.
,
Huang
,
X.
, and
Hua
,
H.
,
2013
, “
On the Characteristics of a Quasi-Zero Stiffness Isolator Using Euler Buckled Beam as Negative Stiffness Corrector
,”
J. Sound Vib.
,
332
(
14
), pp.
3359
3376
.10.1016/j.jsv.2012.10.037
12.
Johnson
,
D. R.
,
Harne
,
R. L.
, and
Wang
,
K. W.
,
2014
, “
A Disturbance Cancellation Perspective on Vibration Control Using a Bistable Snap-Through Attachment
,”
ASME J. Vib. Acoust.
,
136
(
3
), p.
031006
.10.1115/1.4026673
13.
He
,
J. H.
,
2006
, “
Some Asymptotic Methods for Strongly Nonlinear Equations
,”
Int. J. Mod. Phys. B
,
20
(
10
), pp.
1141
1199
.10.1142/S0217979206033796
14.
Kovacic
,
I.
, and
Brennan
,
M. J.
,
2011
,
The Duffing Equation: Nonlinear Oscillators and Their Behaviour
,
Wiley
,
Chichester, UK
.
15.
Peng
,
Z. K.
,
Lang
,
Z. Q.
,
Billings
,
S. A.
, and
Tomlinson
,
G. R.
,
2008
, “
Comparisons Between Harmonic Balance and Nonlinear Output Frequency Response Function in Nonlinear System Analysis
,”
J. Sound Vib.
,
311
(
1–2
), pp.
56
73
.10.1016/j.jsv.2007.08.035
16.
Leung
,
A. Y. T.
,
Guo
,
Z.
, and
Yang
,
H. X.
,
2012
, “
Residue Harmonic Balance Analysis for the Damped Duffing Resonator Driven by a van der Pol Oscillator
,”
Int. J. Mech. Sci.
,
63
(
1
), pp.
59
65
.10.1016/j.ijmecsci.2012.06.011
17.
Dai
,
H. H.
,
Schnoor
,
M.
, and
Atluri
,
S. N.
,
2012
, “
A Simple Collocation Scheme for Obtaining the Periodic Solutions of the Duffing Equation, and Its Equivalence to the High Dimensional Harmonic Balance Method: Subharmonic Oscillations
,”
Comput. Model. Eng. Sci.
,
84
(
5
), pp.
459
497
.10.3970/cmes.2012.084.459
18.
Yuste
,
S. B.
,
1991
, “
Comments on the Method of Harmonic Balance in Which Jacobi Elliptic Functions Are Used
,”
J. Sound Vib.
,
145
(
3
), pp.
381
390
.10.1016/0022-460X(91)90109-W
19.
Yuste
,
S. B.
,
1992
, “
‘Cubication’ of Non-Linear Oscillators Using the Principle of Harmonic Balance
,”
Int. J. Non-Linear Mech.
,
27
(
3
), pp.
347
356
.10.1016/0020-7462(92)90004-Q
20.
Hu
,
H.
, and
Tang
,
J. H.
,
2006
, “
Solution of a Duffing-Harmonic Oscillator by the Method of Harmonic Balance
,”
J. Sound Vib.
,
294
(
3
), pp.
637
639
.10.1016/j.jsv.2005.12.025
21.
Blair
,
K. B.
,
Krousgrill
,
C. M.
, and
Farris
,
T. N.
,
1997
, “
Harmonic Balance and Continuation Techniques in the Dynamic Analysis of Duffing's Equation
,”
J. Sound Vib.
,
202
(
5
), pp.
717
731
.10.1006/jsvi.1996.0863
22.
Yuste
,
S. B.
, and
Bejarano
,
J. D.
,
1986
, “
Construction of Approximate Analytical Solutions to a New Class of Nonlinear Oscillator Equations
,”
J. Sound Vib.
,
110
(
2
), pp.
347
350
.10.1016/S0022-460X(86)80215-2
23.
Gilmore
,
R. J.
, and
Steer
,
M. B.
,
1991
, “
Nonlinear Circuit Analysis Using the Method of Harmonic Balance—A Review of the Art. Part I. Introductory Concepts
,”
Int. J. Microwave Millimeter-Wave Comput.-Aided Eng.
,
1
(
1
), pp.
22
37
.10.1002/mmce.4570010104
24.
Rizzoli
,
V.
,
Mastri
,
F.
, and
Masotti
,
D.
,
1994
, “
General Noise Analysis of Nonlinear Microwave Circuits by the Piecewise Harmonic-Balance Technique
,”
IEEE Trans. Microwave Theory Tech.
,
42
(
5
), pp.
807
819
.10.1109/22.293529
25.
Mickens
,
R. E.
,
1984
, “
Comments on the Method of Harmonic Balance
,”
J. Sound Vib.
,
94
(
3
), pp.
456
460
.10.1016/S0022-460X(84)80025-5
26.
Mickens
,
R. E.
,
1986
, “
A Generalization of the Method of Harmonic Balance
,”
J. Sound Vib.
,
111
(
3
), pp.
515
518
.10.1016/S0022-460X(86)81410-9
27.
Szemplinska-Stupnicka
,
W.
, and
Jerzy
,
R.
,
1993
, “
Steady States in the Twin-Well Potential Oscillator: Computer Simulations and Approximate Analytical Studies
,”
Chaos
,
3
(
3
), pp.
375
385
.10.1063/1.165945
28.
Genesio
,
R.
, and
Tesi
,
A.
,
1992
, “
Harmonic Balance Methods for the Analysis of Chaotic Dynamics in Nonlinear Systems
,”
Automatica
,
28
(
3
), pp.
531
548
.10.1016/0005-1098(92)90177-H
29.
Raghothama
,
A.
, and
Narayanan
,
S.
,
1999
, “
Bifurcation and Chaos in Geared Rotor Bearing System by Incremental Harmonic Balance Method
,”
J. Sound Vib.
,
226
(
3
), pp.
469
492
.10.1006/jsvi.1999.2264
30.
Yasuda
,
K.
,
Kamakura
,
S.
, and
Watanabe
,
K.
,
1988
, “
Identification of Nonlinear Multi-Degree-of-Freedom Systems: Presentation of an Identification Technique
,”
JSME Int. J., Ser. III
,
31
(
1
), pp.
8
14
.http://ci.nii.ac.jp/els/110002504864.pdf?id=ART0002767980&type=pdf&lang=en&host=cinii&order_no=&ppv_type=0&lang_sw=&no=1433450107&cp=
31.
Feeny
,
B. F.
,
Yuan
,
C. M.
, and
Cusumano
,
J. P.
,
2001
, “
Parametric Identification of an Experimental Magneto-Elastic Oscillator
,”
J. Sound Vib.
,
247
(
5
), pp.
785
806
.10.1006/jsvi.2001.3694
32.
Hayashi
,
C.
,
1953
, “
Forced Oscillations With Nonlinear Restoring Force
,”
J. Appl. Phys.
,
24
(
2
), pp.
198
206
.10.1063/1.1721238
33.
Urabe
,
M.
,
1969
, “
Numerical Investigation of Subharmonic Solutions to Duffing's Equation
,”
Publ. Res. Inst. Math. Sci.
,
5
(
1
), pp.
79
112
.10.2977/prims/1195194754
34.
Guckenheimer
,
J.
, and
Holmes
,
P.
,
1983
,
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
,
Springer-Verlag
,
New York
.
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