The characteristics of two, three, and four nonlinear vibration absorbers or nonlinear tuned mass dampers (NTMDs) attached to a structure under harmonic excitation are investigated. The frequency response curves are theoretically determined using van der Pol’s method. When the parameters of the absorbers are equal, it is found from the theoretical analysis that pitchfork bifurcations may occur on the part of the response curves, which are unstable in the multi-absorber systems, but are stable in a system with one NTMD. Multivalued steady-state solutions, such as three steady-state solutions for a dual-absorber system with different amplitudes, five steady-state solutions for a triple-absorber system, and seven steady-state solutions for a quadruple-absorber system, appear near bifurcation points. The NTMDs behave in that one of them vibrates at high amplitudes while the others vibrate at low amplitudes, even if the dimensions of the NTMDs are identical. Namely, “localization phenomenon” or “mode localization” occurs. After the pitchfork bifurcation, Hopf bifurcations may occur depending on the values of the system parameters, and amplitude- and phase-modulated motions, including chaotic vibrations, appear after the Hopf bifurcation when the excitation frequency decreases. Lyapunov exponents are numerically calculated to prove the occurrence of chaotic vibrations. Bifurcation sets are also calculated to investigate the influence of the system parameters on the response of the systems.
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April 2010
Research Papers
Bifurcation Phenomena Caused by Multiple Nonlinear Vibration Absorbers
Takashi Ikeda
Takashi Ikeda
Department of Mechanical System Engineering, Graduate School of Engineering,
e-mail: tikeda@hiroshima-u.ac.jp
Hiroshima University
, 1-4-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8527, Japan
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Takashi Ikeda
Department of Mechanical System Engineering, Graduate School of Engineering,
Hiroshima University
, 1-4-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8527, Japane-mail: tikeda@hiroshima-u.ac.jp
J. Comput. Nonlinear Dynam. Apr 2010, 5(2): 021012 (15 pages)
Published Online: March 2, 2010
Article history
Received:
January 5, 2009
Revised:
April 27, 2009
Published:
March 2, 2010
Citation
Ikeda, T. (March 2, 2010). "Bifurcation Phenomena Caused by Multiple Nonlinear Vibration Absorbers." ASME. J. Comput. Nonlinear Dynam. April 2010; 5(2): 021012. https://doi.org/10.1115/1.4000790
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