We have studied a nonlinear spring-mass chain loaded by a quasistatic pull. The spring forces are assumed to be cubic with intervals of negative stiffness. Depending on the parameters, the system has multiple equilibria. The normal form and the bifurcation behaviors for the single- and two-degree-of-freedom systems are studied in detail. A new type of bifurcation, which we refer to as a star bifurcation, has been observed for the symmetric two-degree-of-freedom system. This bifurcation is of codimension-four for the undamped case and codimension-three or two for the damped case, depending on the form of the damping.

References

References
1.
Feeny
,
B. F.
, and
Diaz
,
A. R.
,
2010
, “
Twinkling Phenomena in Snap-Through Oscillators
,”
ASME J. Vib. Acoust.
,
132
(
6
), p.
061013
.10.1115/1.4000764
2.
Wang
,
Y. C.
, and
Lakes
,
R. S.
,
2004
, “
Extreme Stiffness Systems Due to Negative Stiffness Elements
,”
Am. J. Phys.
,
72
, pp. 40–50.10.1119/1.1619140
3.
Puglisi
,
G.
, and
Truskinovsky
,
L.
,
2000
, “
Mechanics of a Discrete Chain With Bi-Stable Elements
,”
J. Mech. Phys. Solid.
,
48
(
1
), pp.
1
27
.10.1016/S0022-5096(99)00006-X
4.
Puglisi
,
G.
, and
Truskinovsky
,
L.
,
2002
, “
Rate Independent Hysteresis in a Bi-Stable Chain
,”
J. Mech. Phys. Solid.
,
50
(
2
), pp.
165
187
.10.1016/S0022-5096(01)00055-2
5.
Puglisi
,
G.
,
2006
, “
Hysteresis in Multi-Stable Lattices With Non-Local Interactions
,”
J. Mech. Phys. Solid.
,
54
(
10
), pp.
2060
2088
.10.1016/j.jmps.2006.04.006
6.
Sarafian
,
H.
,
2010
, “
Static Electric-Spring and Nonlinear Oscillations
,”
JEMAA
,
2
(
2
), pp.
75
81
.10.4236/jemaa.2010.22011
7.
Sarafian
,
H.
,
2011
, “
Nonlinear Oscillations of a Magneto Static Spring-Mass
,”
JEMAA
,
3
, pp.
133
139
.10.4236/jemaa.2011.35022
8.
Roundy
,
S.
, and
Wright
,
P. K.
,
2004
, “
A Piezoelectric Vibration Based Generator for Wireless Electronics
,”
Smart Mater. Struct.
,
13
, pp. 1131–1142.10.1088/0964-1726/13/5/018
9.
Sodano
,
H. A.
,
Inman
,
D. J.
, and
Park
,
G.
,
2005
, “
Comparison of Piezoelectric Energy Harvesting Devices for Recharging Batteries
,”
J. Intel. Mater. Syst. Struct.
,
16
(
10
), pp. 799–807.10.1177/1045389X05056681
10.
Liao
,
Y.
, and
Sodano
,
H. A.
,
2008
, “
Model of a Single Mode Energy Harvester and Properties for Optimal Power Generation
,”
Smart Mater. Struct.
,
17
, p.
065026
.10.1088/0964-1726/17/6/065026
11.
Renno
,
J. M.
,
Daqaq
,
M. F.
, and
Inman
,
D. J.
,
2009
, “
On the Optimal Energy Harvesting From a Vibration Source
,”
J. Sound Vib.
,
320
(
1–2
), pp.
386
405
.10.1016/j.jsv.2008.07.029
12.
Scruggs
,
J. T.
, and
Behrens
,
S.
,
2011
, “
Optimal Energy Harvesting From Low-Frequency Bistate Force Loadings
,”
ASME J. Vib. Acoust.
,
133
, p.
011008
.10.1115/1.4002792
13.
Anton
,
S. R.
, and
Sodano
,
H. A.
,
2007
, “
A Review of Power Harvesting Using Piezoelectric Materials (2003–2006)
,”
Smart Mater. Struct.
,
16
, p.
R1
.10.1088/0964-1726/16/3/R01
14.
Stanton
,
S. C.
,
Erturk
,
A.
,
Mann
,
B. P.
, and
Inman
,
D. J.
,
2010
, “
Nonlinear Piezoelectricity in Electroelastic Energy Harvesters: Modeling and Experimental Identification
,”
J. Appl. Phys.
,
108
(
7
), p.
074903
.10.1063/1.3486519
15.
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 Nonlinear Phenom.
,
239
(
10
), pp.
640
653
.10.1016/j.physd.2010.01.019
16.
Kim
,
S.
,
Clark
,
W. W.
, and
Wang
,
Q. M.
,
2005
, “
Piezoelectric Energy Harvesting With a Clamped Circular Plate: Experimental Study
,”
J. Intel. Mater. Syst. Struct.
,
16
(
10
), pp. 855–863.10.1177/1045389X05054043
17.
Mo
,
C.
,
Radziemski
,
L. J.
, and
Clark
,
W. W.
,
2010
, “
Experimental Validation of Energy Harvesting Performance for Pressure-Loaded Piezoelectric Circular Diaphragms
,”
Smart Mater. Struct.
,
19
, p.
075010
.10.1088/0964-1726/19/7/075010
18.
Kerschen
,
G.
,
McFarland
,
D. M.
,
Kowtko
,
J. J.
,
Lee
,
Y. S.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2007
, “
Experimental Demonstration of Transient Resonance Capture in a System of Two Coupled Oscillators With Essential Stiffness Nonlinearity
,”
J. Sound Vib.
,
299
(
4–5
), pp.
822
838
.10.1016/j.jsv.2006.07.029
19.
Quinn
,
D. D.
,
Triplett
,
A. L.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2011
, “
Comparing Linear and Essentially Nonlinear Vibration-Based Energy Harvesting
,”
ASME J. Vib. Acoust.
,
133
(1), p.
011001
.10.1115/1.4002782
20.
Quinn
,
D. D.
,
Triplett
,
A. L.
,
Vakakis
,
A. F.
, and
Bergman
,
L. A.
,
2011
, “
Energy Harvesting From Impulsive Loads Using Intentional Essential Nonlinearities
,”
ASME J. Vib. Acoust.
,
133
(1), p.
011004
.10.1115/1.4002787
21.
McFarland
,
D. M.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2005
, “
Experimental Study of Non-Linear Energy Pumping Occurring at a Single Fast Frequency
,”
Int. J. Nonlinear Mech.
,
40
(
6
), pp.
891
899
.10.1016/j.ijnonlinmec.2004.11.001
22.
Gourdon
,
E.
,
Alexander
,
N. A.
,
Taylor
,
C. A.
,
Lamarque
,
C. H.
, and
Pernot
,
S.
,
2007
, “
Nonlinear Energy Pumping Under Transient Forcing With Strongly Nonlinear Coupling: Theoretical and Experimental Results
,”
J. Sound Vib.
,
300
(
3–5
), pp.
522
551
.10.1016/j.jsv.2006.06.074
23.
Manevitch
,
L. I.
,
Musienko
,
A. I.
, and
Lamarque
,
C. H.
,
2007
, “
New Analytical Approach to Energy Pumping Problem in Strongly Nonhomogeneous 2DOF Systems
,”
Meccanica
,
42
(
1
), pp.
77
83
.10.1007/s11012-006-9021-y
24.
Tsakirtzis
,
S.
,
Panagopoulos
,
P. N.
,
Kerschen
,
G.
,
Gendelman
,
O.
,
Vakakis
,
A. F.
, and
Bergman
,
L. A.
,
2007
, “
Complex Dynamics and Targeted Energy Transfer in Linear Oscillators Coupled to Multi-Degree-of-Freedom Essentially Nonlinear Attachments
,”
Nonlinear Dyn.
,
48
(
3
), pp.
285
318
.10.1007/s11071-006-9089-x
25.
Quinn
,
D. D.
,
Gendelman
,
O.
,
Kerschen
,
G.
,
Sapsis
,
T. P.
,
Bergman
,
L. A.
, and
Vakakis
,
A. F.
,
2008
, “
Efficiency of Targeted Energy Transfers in Coupled Nonlinear Oscillators Associated With 1:1 Resonance Captures: Part I
,”
J. Sound Vib.
,
311
(
3–5
), pp.
1228
1248
.10.1016/j.jsv.2007.10.026
26.
Sapsis
,
T. P.
Vakakis
,
A. F.
,
Gendelman
,
O. V.
,
Bergman
,
L. A.
,
Kerschen
,
G.
, and
Quinn
,
D. D.
,
2009
, “
Efficiency of Targeted Energy Transfers in Coupled Nonlinear Oscillators Associated With 1:1 Resonance Captures: Part II, Analytical Study
,”
J. Sound Vib.
,
325
(
1–2
), pp.
297
320
.10.1016/j.jsv.2009.03.004
27.
Vakakis
,
A.
,
Gendelman
,
O.
,
Bergman
,
L.
,
McFarland
,
D.
,
Kerschen
,
G.
, and
Lee
,
Y. S.
,
2009
,
Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems: I and II
,
Springer
, Berlin.
28.
Bellet
,
R.
,
Cochelin
,
B.
,
Herzog
,
P.
, and
Mattei
,
P. O.
,
2010
, “
Experimental Study of Targeted Energy Transfer From an Acoustic System to a Nonlinear Membrane Absorber
,”
J. Sound Vib.
,
329
(
14
), pp.
2768
2791
.10.1016/j.jsv.2010.01.029
29.
Priya
,
S.
, and
Inman
,
D. J.
,
2008
,
Energy Harvesting Technologies
,
Springer
, New York.
30.
Stephen
,
N. G.
,
2006
, “
On Energy Harvesting From Ambient Vibration
,”
J. Sound Vib.
,
293
(
1–2
), pp.
409
425
.10.1016/j.jsv.2005.10.003
31.
Mann
,
B. P.
, and
Sims
,
N. D.
,
2009
, “
Energy Harvesting From the Nonlinear Oscillations of Magnetic Levitation
,”
J. Sound Vib.
,
319
(
1–2
), pp.
515
530
.10.1016/j.jsv.2008.06.011
32.
Kurs
,
A.
,
Karalis
,
A.
,
Moffatt
,
R.
,
Joannopoulos
,
J. D.
,
Fisher
,
P.
, and
Soljačić
,
M.
,
2007
, “
Wireless Power Transfer Via Strongly Coupled Magnetic Resonances
,”
Science
,
317
(
5834
), pp. 83–86.10.1126/science.1143254
33.
Karalis
,
A.
,
Joannopoulos
,
J. D.
, and
Soljačić
,
M.
,
2008
, “
Efficient Wireless Non-Radiative Mid-Range Energy Transfer
,”
Ann. Phys.
,
323
(
1
), pp.
34
48
.10.1016/j.aop.2007.04.017
34.
Wei
,
X. C.
,
Li
,
E. P.
,
Guan
,
Y. L.
, and
Chong
,
Y. H.
,
2009
, “
Simulation and Experimental Comparison of Different Coupling Mechanisms for the Wireless Electricity Transfer
,”
J. Electromag. Waves Appl.
,
23
(
7
), pp.
925
934
.10.1163/156939309788355180
35.
Kurs
,
A.
,
Moffatt
,
R.
, and
Soljačić
,
M.
,
2010
, “
Simultaneous Mid-Range Power Transfer to Multiple Devices
,”
Appl. Phys. Lett.
,
96
(
4
), p.
044102
.10.1063/1.3284651
36.
Peng
,
L.
,
Breinbjerg
,
O.
, and
Mortensen
,
N. A.
,
2010
, “
Wireless Energy Transfer Through Non-Resonant Magnetic Coupling
,”
J. Electromag. Waves Appl.
,
24
(
11–12
), pp.
1587
1598
.10.1163/156939310792149795
37.
Peng
,
L.
,
Wang
,
J. Y.
,
Ran
,
L. X.
,
Breinbjerg
,
O.
, and
Mortensen
,
N. A.
,
2011
, “
Performance Analysis and Experimental Verification of Mid-Range Wireless Energy Transfer Through Non-Resonant Magnetic Coupling
,”
J. Electromag. Waves Appl.
,
25, 5
(
6
), pp.
845
855
.10.1163/156939311794827186
38.
Tesla
,
N.
1914
, “
Apparatus for Transmitting Electrical Energy
,”
U.S. Patent No. 1,119,73
2
.
39.
Panigrahi
,
S. R.
,
Feeny
,
B. F.
, and
Diaz
,
A. R.
,
2012
, “
Bifurcations in Twinkling Oscillators
,”
Proceedings of the ASME International Design Engineering Technical Conferences
, 24th Conference on Mechanical Vibration and Noise, Chicago, IL, August 12–15,
ASME
Paper No. DETC2012-70943.10.1115/DETC2012-70943
40.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
,
1995
,
Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods
,
Wiley
, New York.
41.
Wiggins
,
S.
,
2003
,
Introduction to Applied Nonlinear Dynamical Systems and Chaos
, Vol.
2
,
Springer
, New York.
42.
Panigrahi
,
S. R.
,
Feeny
,
B. F.
, and
Diaz
,
A. R.
,
2013
, “
`Eclipse’ Bifurcation in a Twinkling Oscillator
,”
ASME J. Vib. Acoust.
(accepted).
43.
Panigrahi
,
S. R.
,
Feeny
,
B. F.
, and
Diaz
,
A. R.
,
2013
, “
Second-Order Perturbation Analysis of Low Amplitude Traveling Waves in a Periodic Nonlinear Chain
,”
Proceedings of the ASME International Design Engineering Technical Conferences
, 25th Conference on Mechanical Vibration and Noise, Portland, OR, August 4–7, ASME Paper No. DETC2013/13207.
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