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.
Issue Section:
Research Papers
References
1.
Feeny
, B. F.
, and Diaz
, A. R.
, 2010
, “Twinkling Phenomena in Snap-Through Oscillators
,” ASME J. Vib. Acoust.
, 132
(6
), p. 061013
.10.1115/1.40007642.
Wang
, Y. C.
, and Lakes
, R. S.
, 2004
, “Extreme Stiffness Systems Due to Negative Stiffness Elements
,” Am. J. Phys.
, 72
, pp. 40–50.10.1119/1.16191403.
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-X4.
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)00055-25.
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.2006.04.0066.
Sarafian
, H.
, 2010
, “Static Electric-Spring and Nonlinear Oscillations
,” JEMAA
, 2
(2
), pp. 75
–81
.10.4236/jemaa.2010.220117.
Sarafian
, H.
, 2011
, “Nonlinear Oscillations of a Magneto Static Spring-Mass
,” JEMAA
, 3
, pp. 133
–139
.10.4236/jemaa.2011.350228.
Roundy
, S.
, and Wright
, P. K.
, 2004
, “A Piezoelectric Vibration Based Generator for Wireless Electronics
,” Smart Mater. Struct.
, 13
, pp. 1131–1142.10.1088/0964-1726/13/5/0189.
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. 799–807.10.1177/1045389X0505668110.
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
.10.1088/0964-1726/17/6/06502611.
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.2008.07.02912.
Scruggs
, J. T.
, and Behrens
, S.
, 2011
, “Optimal Energy Harvesting From Low-Frequency Bistate Force Loadings
,” ASME J. Vib. Acoust.
, 133
, p. 011008
.10.1115/1.400279213.
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/R0114.
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
.10.1063/1.348651915.
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.2010.01.01916.
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. 855–863.10.1177/1045389X0505404317.
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
.10.1088/0964-1726/19/7/07501018.
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.2006.07.02919.
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
.10.1115/1.400278220.
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
.10.1115/1.400278721.
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.2004.11.00122.
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.2006.06.07423.
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-y24.
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-x25.
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.2007.10.02626.
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.2009.03.00427.
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.2010.01.02929.
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.2005.10.00331.
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.2008.06.01132.
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.114325433.
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.2007.04.01734.
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
.10.1163/15693930978835518035.
Kurs
, A.
, Moffatt
, R.
, and Soljačić
, M.
, 2010
, “Simultaneous Mid-Range Power Transfer to Multiple Devices
,” Appl. Phys. Lett.
, 96
(4
), p. 044102
.10.1063/1.328465136.
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
.10.1163/15693931079214979537.
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
.10.1163/15693931179482718638.
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/DETC2012-7094340.
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.Copyright © 2014 by ASME
You do not currently have access to this content.
