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TECHNICAL PAPERS

Isolated Resonance Captures and Resonance Capture Cascades Leading to Single- or Multi-Mode Passive Energy Pumping in Damped Coupled Oscillators

[+] Author and Article Information
Alexander F. Vakakis

Division of Mechanics, National Technical University of AthensDepartment of Mechanical and Industrial Engineering (adjunct), University of Illinois at Urbana-Champaign, Urbana, IL

D. Michael McFarland, Lawrence Bergman

Dept. of Aeronautical and Astronautical Engineering, University of Illinois at Urbana-Champaign, Urbana, IL

Leonid I. Manevitch, Oleg Gendelman

Institute of Chemical Physics, Russian Academy of Sciences, Moscow

J. Vib. Acoust 126(2), 235-244 (May 04, 2004) (10 pages) doi:10.1115/1.1687397 History: Received July 01, 2002; Revised August 01, 2003; Online May 04, 2004
Copyright © 2004 by ASME
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References

Gendelman,  O., 2001, “Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators,” Nonlinear Dyn., 25, pp. 237–253.
Vakakis,  A. F., 2001, “Inducing Passive Nonlinear Energy Sinks in Linear Vibrating Systems,” ASME J. Vibr. Acoust., 123(3), pp. 324–332.
Arnold, V. I., ed., 1988, Dynamical Systems III, Encyclopaedia of Mathematical Sciences Vol. 3, Springer Verlag, Berlin and New York.
Vakakis,  A. F., Manevitch,  L. I., Gendelman,  Q., and Bergman,  L., 2003, “Dynamics of Linear Discrete Systems Connected to Local Essentially Nonlinear Attachments,” J. Sound Vib., 264, pp. 559-577.
Vakakis, A. F., Manevitch, L. I., Mikhlin, Yu. V., Pilipchuk, V. N., and Zevin, A. A., 1996, Normal Modes and Localization in Nonlinear Systems, Wiley Interscience, New York.
Vakakis,  A. F., 2003, “Designing a Linear Structure with a Local Nonlinear Attachment For Enhanced Energy Pumping,” Meccanica, 38(6), pp. 677–686.
Shaw,  S., and Pierre,  C., 1991, “Nonlinear Normal Modes and Invariant Manifolds,” J. Sound Vib., 150(1), pp. 170–173.
Shaw,  S., and Pierre,  C., 1993, “Normal Modes for Nonlinear Vibratory Systems,” J. Sound Vib., 164(1), pp. 85–124.
Vakakis,  A. F., and Gendelman,  O., 2001, “Energy Pumping in Nonlinear Mechanical Oscillators II: Resonance capture,” ASME J. Appl. Mech., 68(1), pp. 42–48.
Wiggins, S., 1989, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer Verlag, Berlin and New York.

Figures

Grahic Jump Location
The three-degree-of-freedom system with a NES attachment
Grahic Jump Location
Schematic of the synthesized NNM frequencies as functions of the (conserved) energy of the combined system: –Stable, ×××× unstable NNMs. With dashed lines we depict the “backbone curves” of the uncoupled system corresponding to ε=0. Schematics of the displacements of the particles of the system for each NNM are also depicted 4.
Grahic Jump Location
Initial excitation of linear out-of-phase mode, Y=2.7: (a) Instantaneous frequency of the NES; (b) Responses [[dashed_line]]y0(t) and –y1(t); (c) Responses [[dashed_line]]y0(t) and –v(t)
Grahic Jump Location
Initial excitation of linear out-of-phase mode, Y=2.8: (a) Instantaneous frequency of the NES; (b) Responses [[dashed_line]]y0(t) and –y1(t); (c) Responses [[dashed_line]]y0(t) and –v(t)
Grahic Jump Location
Initial excitation of linear in-phase mode, Y=1.0: (a) Instantaneous frequency of the NES; (b) Responses [[dashed_line]]y0(t) and –y1(t); (c) Responses [[dashed_line]]y0(t) and –v(t)
Grahic Jump Location
Transient excitation (F0=160) of the a symmetric three-DOF system: (a) Instantaneous frequency of the NES, with the levels corresponding to the two eigenfrequencies of the linear subsystem indicated by dashed lines; (b) Responses [[dashed_line]]y0(t) and –v(t); (c) Responses [[dashed_line]]y0(t) and –y1(t); (d) Portion of external energy absorbed and dissipated by the NES; (e) Comparison between the linear [[dashed_line]]and nonlinear –responses for y1(t)
Grahic Jump Location
Portion of total energy dissipated at the NES for decreasing strengths of the input force (F0≤160); In each case the instantaneous frequency of the NES is plotted versus time
Grahic Jump Location
Portion of total energy dissipated at the NES for increasing strengths of the input force (F0≥160); In each case the instantaneous frequency of the NES is plotted versus time

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