The excitation-induced stability (EIS) phenomenon in a harmonically excited bistable Duffing oscillator is studied in this paper. Criteria to predict system and excitation conditions necessary to maintain EIS are derived through a combination of the method of harmonic balance, perturbation theory, and stability theory for Mathieu's equation. Accuracy of the criteria is verified by analytical and numerical studies. We demonstrate that damping primarily determines the likelihood of attaining EIS response when several dynamics coexist while excitation level governs both the existence and frequency range of the EIS region, providing comprehensive guidance for realizing or avoiding EIS dynamics. Experimental results are in good agreement regarding the comprehensive influence of excitation conditions on the inducement of EIS.

References

References
1.
Bolotin
,
V. V.
,
1964
,
The Dynamic Stability of Elastic Systems
,
Holden Day
,
San Francisco
.
2.
Holmes
,
P.
,
1979
, “
A Nonlinear Oscillator With a Strange Attractor
,”
Philos. Trans. R. Soc. London
,
292
(
1394
), pp.
419
448
.10.1098/rsta.1979.0068
3.
Moon
,
F. C.
, and
Holmes
,
P. J.
,
1979
, “
A Magnetoelastic Strange Attractor
,”
J. Sound Vib.
,
65
(
2
), pp.
275
296
.10.1016/0022-460X(79)90520-0
4.
Moon
,
F. C.
,
1980
, “
Experiments on Chaotic Motions of a Forced Nonlinear Oscillator: Strange Attractors
,”
ASME J. Appl. Mech.
,
47
(
3
), pp.
638
644
.10.1115/1.3153746
5.
Dowell
,
E. H.
, and
Pezeshki
,
C.
,
1986
, “
On the Understanding of Chaos in Duffings Equation Including a Comparison With Experiment
,”
ASME J. Appl. Mech.
,
53
(
1
), pp.
5
9
.10.1115/1.3171739
6.
Tang
,
D. M.
, and
Dowell
,
E. H.
,
1988
, “
On the Threshold Force for Chaotic Motions for a Forced Buckled Beam
,”
ASME J. Appl. Mech.
,
55
(
1
), pp.
190
196
.10.1115/1.3173628
7.
Szemplińska-Stupnicka
,
W.
, and
Rudowski
,
J.
,
1992
, “
Local Methods in Predicting Occurrence of Chaos in Two-Well Potential Systems: Superharmonic Frequency Region
,”
J. Sound Vib.
,
152
(
1
), pp.
57
72
.10.1016/0022-460X(92)90065-6
8.
Stanton
,
S. C.
,
Mann
,
B. P.
, and
Owens
,
B. A. M.
,
2012
, “
Melnikov Theoretic Methods for Characterizing the Dynamics of the Bistable Piezoelectric Inertial Generator in Complex Spectral Environments
,”
Physica D
,
241
(
6
), pp.
711
720
.10.1016/j.physd.2011.12.010
9.
Tseng
,
W.-Y.
, and
Dugundji
,
J.
,
1971
, “
Nonlinear Vibrations of a Buckled Beam Under Harmonic Excitation
,”
ASME J. Appl. Mech.
,
38
(
2
), pp.
467
476
.10.1115/1.3408799
10.
Szemplińska-Stupnicka
,
W.
, and
Rudowski
,
J.
,
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
11.
Harne
,
R. L.
,
Thota
,
M.
, and
Wang
,
K. W.
,
2013
, “
Concise and High-Fidelity Predictive Criteria for Maximizing Performance and Robustness of Bistable Energy Harvesters
,”
Appl. Phys. Lett.
,
102
(
5
), p.
053903
.10.1063/1.4790381
12.
Ibrahim
,
R. A.
,
2006
, “
Excitation-Induced Stability and Phase Transition: A Review
,”
J. Vib. Control
,
12
(
10
), pp.
1093
1170
.10.1177/1077546306069912
13.
Blackburn
,
J. A.
,
Smith
,
H. J. T.
, and
Grønbech-Jensen
,
N.
,
1992
, “
Stability and Hopf Bifurcations in an Inverted Pendulum
,”
Am. J. Phys.
,
60
(
10
), pp.
903
908
.10.1119/1.17011
14.
Butikov
,
E. I.
,
2001
, “
On the Dynamic Stabilization of an Inverted Pendulum
,”
Am. J. Phys.
,
69
(
7
), pp.
755
768
.10.1119/1.1365403
15.
Seyranian
,
A. A.
, and
Seyranian
,
A. P.
,
2006
, “
The Stability of an Inverted Pendulum With a Vibrating Suspension Point
,”
J. Appl. Math. Mech.
,
70
(
5
), pp.
754
761
.10.1016/j.jappmathmech.2006.11.009
16.
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
17.
Kim
,
S.-Y.
, and
Kim
,
Y.
,
2000
, “
Dynamic Stabilization in the Double-Well Duffing Oscillator
,”
Phys. Rev. E
,
61
(
6
), pp.
6517
6520
.10.1103/PhysRevE.61.6517
18.
Kim
,
Y.
,
Lee
,
S. Y.
, and
Kim
,
S.-Y.
,
2000
, “
Experimental Observation of Dynamic Stabilization in a Double-Well Duffing Oscillator
,”
Phys. Lett. A
,
275
(
4
), pp.
254
259
.10.1016/S0375-9601(00)00572-7
19.
Karami
,
M. A.
, and
Inman
,
D. J.
,
2011
, “
Equivalent Damping and Frequency Change for Linear and Nonlinear Hybrid Vibrational Energy Harvesting Systems
,”
J. Sound Vib.
,
330
(
23
), pp.
5583
5597
.10.1016/j.jsv.2011.06.021
20.
Wu
,
Z.
,
Harne
,
R. L.
, and
Wang
,
K. W.
,
2014
, “
Energy Harvester Synthesis Via Coupled Linear-Bistable System With Multistable Dynamics
,”
ASME J. Appl. Mech.
,
81
(
6
), p.
061005
.10.1115/1.4026555
21.
Johnson
,
D. R.
,
Harne
,
R. L.
, and
Wang
,
K. W.
,
2013
, “
A Disturbance Cancellation Perspective on Vibration Control Using a Bistable Snap Through Attachment
,”
ASME J. Vib. Acoust.
(accepted).10.1115/1.4026673
22.
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
23.
Masana
,
R.
, and
Daqaq
,
M. F.
,
2011
, “
Relative Performance of a Vibratory Energy Harvester in Mono- and Bi-Stable Potentials
,”
J. Sound Vib.
,
330
(
24
), pp.
6036
6052
.10.1016/j.jsv.2011.07.031
24.
Ibrahim
,
R. A.
,
2008
, “
Recent Advances in Nonlinear Passive Vibration Isolators
,”
J. Sound Vib.
,
314
(
3–5
), pp.
371
452
.10.1016/j.jsv.2008.01.014
25.
Friswell
,
M. I.
, and
Penny
,
J. E. T.
,
1994
, “
The Accuracy of Jump Frequencies in Series Solutions of the Response of a Duffing Oscillator
,”
J. Sound Vib.
,
169
(
2
), pp.
261
269
.10.1006/jsvi.1994.1018
26.
Stanton
,
S. C.
,
Owens
,
B. A. M.
, and
Mann
,
B. P.
,
2012
, “
Harmonic Balance Analysis of the Bistable Piezoelectric Inertial Generator
,”
J. Sound Vib.
,
331
(
15
), pp.
3617
3627
.10.1016/j.jsv.2012.03.012
27.
Gunderson
,
H.
,
Rigas
,
H.
, and
VanVleck
,
F. S.
,
1974
, “
A Technique for Determining Stability Regions for the Damped Mathieu Equation
,”
SIAM J. Appl. Math.
,
26
(
2
), pp.
345
349
.10.1137/0126032
28.
Jordan
,
D. W.
, and
Smith
,
P.
,
2007
,
Nonlinear Ordinary Differential Equations
,
4th ed.
,
Oxford University Press
,
New York
.
29.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1979
,
Nonlinear Oscillations
,
John Wiley
,
New York
.
You do not currently have access to this content.