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TECHNICAL PAPERS

Energy Pumping in Nonlinear Mechanical Oscillators: Part I—Dynamics of the Underlying Hamiltonian Systems

[+] Author and Article Information
O. Gendelman, L. I. Manevitch

Institute of Chemical Physics, Russian Academy of Sciences, Kosygin Str. 4, 117977 Moscow, Russia

A. F. Vakakis

Department of Mechanical and Industrial Engineering, University of Illinois, 1206 W. Green Street, Urbana, IL 61801 e-mail: avakakis@uiuc.edu

R. M’Closkey

Department of Mechanical and Aerospace Engineering, University of California, 38-137N Engineering IV, 405 Hilgard Avenue, Los Angeles, CA 90024-1597 e-mail: obsidian.seas.ucla.edu

J. Appl. Mech 68(1), 34-41 (May 02, 2000) (8 pages) doi:10.1115/1.1345524 History: Received September 29, 1999; Revised May 02, 2000
Copyright © 2001 by ASME
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References

Vakakis,  A. F., and Gendelman,  O., 2001, “Energy ‘Pumping’ in Coupled Mechanical Oscillators II: Resonance Capture,” ASME J. Appl. Mech., 68, pp. 42–48.
Hodges,  C. H., 1982, “Confinement of Vibration by Structural Irregularity,” J. Sound Vib., 82, No. 3, pp. 411–424.
Pierre,  C., and Dowell,  E. H., 1987, “Localization of Vibrations by Structural Irregularity,” J. Sound Vib., 114, No. 3, pp. 549–564.
Bendiksen,  O. O., 1987, “Mode Localization Phenomena in Large Space Structures,” AIAA J., 25, No. 9, pp. 1241–1248.
Pierre,  C., and Cha,  P., 1989, “Strong Mode Localization in Nearly Periodic Disordered Structures,” AIAA J., 27, No. 2, pp. 227–241.
Photiadis,  D. M., 1992, “Anderson Localization of One-Dimensional Wave Propagation on a Fluid-Loaded Plate,” J. Acoust. Soc. Am., 91, No. 2, pp. 771–780.
Vakakis,  A. F., and Cetinkaya,  C., 1993, “Mode Localization in a Class of Multi-Degree-of-Freedom Nonlinear Systems With Cyclic Symmetry,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 53, pp. 265–282.
Vakakis,  A. F., Nayfeh,  T. A., and King,  M. E., 1993, “A Multiple-Scales Analysis of Nonlinear Localized Modes in a Cyclic Periodic System,” ASME J. Appl. Mech., 60, No. 2, pp. 388–397.
Vakakis, A. F., Manevitch, L. I., Mikhlin, Yu., Pilipchuck, V., and Zevin, A., 1996, Normal Modes and Localization in Nonlinear Systems, John Wiley and Sons, New York.
Nayfeh, A. H., and Mook, D., 1984, Nonlinear Oscillations, John Wiley and Sons, New York.
Nayfeh,  S. A., and Nayfeh,  A. H., 1994, “Energy Transfer From High to Low-Frequency Modes in a Flexible Structure via Modulation,” ASME J. Vibr. Acoust., 116, pp. 203–207.
Gendelman, O., 1999, “Transition of Energy to Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators,” Nonlinear Dyn., submitted for publication.
Holmes,  P. J., and Marsden,  J. E., 1982, “Horseshoes in Perturbations of Hamiltonian Systems With Two Degrees of Freedom,” Commun. Math. Phys., 82, pp. 523–544.
Pilipchuck,  V. N., 1985, “The Calculation of Strongly Nonlinear Systems Close to Vibration-Impact Systems,” Prikl. Mat. Meck. (PMM), 49, No. 6, pp. 572–578.
Pilipchuck,  V. N., 1988, “A Transformation for Vibrating Systems Based on a Non-Smooth Periodic Pair of Functions,” Dokl. Akad. Nauk SSSR, Ser. A, 4, pp. 37–40 (in Russian).
Pilipchuck,  V. N., Vakakis,  A. F., and Azeez,  M. A. F., 1997, “Study of a Class of Subharmonic Motions Using a Non-Smooth Temporal Transformation,” Physica D, 100, pp. 145–164.
Vakakis,  A. F., and Atanackovic,  T. M., 1999, “Buckling of an Elastic Ring Forced by a Periodic Array of Compressive Loads,” ASME J. Appl. Mech., 66, pp. 361–367.

Figures

Grahic Jump Location
Numerical transient responses y1(t) and y2(t) of system (1) for (a) h=0.5, (b) h=0.8, (c) h=1.125; – oscillator 1, [[dashed_line]] oscillator 2
Grahic Jump Location
Numerical transient response of the three-degrees-of-freedom system for (a) ẏ3(0)=2.0, (b) ẏ3(0)=3.0, and (c) ẏ3(0)=4.0; – oscillator 1, [[dotted_line]] oscillator 2, [[dashed_line]] oscillator 3
Grahic Jump Location
Construction of the solution y1 over an entire normalized period (equal to 4) from the half-normalized period solutions (a) eY(τ), and (b) X(τ)
Grahic Jump Location
Leading 2m:1 subharmonic orbits as functions of h: –Y(−1) for m=1, –Y(−1) for m=2, [[dashed_line]]Y(−1) for m=3  
Grahic Jump Location
Leading (2n−1):1 subharmonic orbits as functions of h: –X(−1) for n=1, – −X(−1) for n=2, [[dashed_line]]X(−1) for n=3; ×××× unstable 1:1 subharmonic orbits
Grahic Jump Location
Poincaré maps of the dynamics of the undamped system (1) at varying energy levels for ω22=0.9,C=5.0, ε=0.1: (a) h=0.05, (b) h=0.2, (c) h=0.8, (d) h=2.0
Grahic Jump Location
Phase plots of the system of Eqs. (18) for ω22=0.9,C=5.0, ε=0.1, and varying values of the first integral of motion N: (a) N=0.4, (b) N=0.8, (c) N=1.9

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