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Effect of 1:3 resonance on the steady-state dynamics of a forced strongly nonlinear oscillator with a linear light attachment

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Abstract

We study the 1:3 resonant dynamics of a two degree-of-freedom (DOF) dissipative forced strongly nonlinear system by first examining the periodic steady-state solutions of the underlying Hamiltonian system and then the forced and damped configuration. Specifically, we analyze the steady periodic responses of the two DOF system consisting of a grounded strongly nonlinear oscillator with harmonic excitation coupled to a light linear attachment under condition of 1:3 resonance. This system is particularly interesting since it possesses two basic linearized eigenfrequencies in the ratio 3:1, which, under condition of resonance, causes the localization of the fundamental and third-harmonic components of the responses of the grounded nonlinear oscillator and the light linear attachment, respectively. We examine in detail the topological structure of the periodic responses in the frequency–energy domain by computing forced frequency–energy plots (FEPs) in order to deduce the effects of the 1:3 resonance. We perform complexification/averaging analysis and develop analytical approximations for strongly nonlinear steady-state responses, which agree well with direct numerical simulations. In addition, we investigate the effect of the forcing on the 1:3 resonance phenomena and conclude our study with the stability analysis of the steady-state solutions around 1:3 internal resonance, and a discussion of the practical applications of our findings in the area of nonlinear targeted energy transfer.

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Abbreviations

DOF:

Degree of freedom

FEP:

Frequency–energy plot

NNM:

Nonlinear normal mode

CX-A:

Complexification-averaging method

S11:

Symmetric 1:1 resonance branch

S13:

Symmetric 1:3 resonance branch

TET:

Targeted energy transfer

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Authors and Affiliations

  1. Department of Mechanical Science and Engineering, University of Illinois at Urbana–Champaign, Urbana, IL, 61801, USA

    Mehmet Kurt & Alexander F. Vakakis

  2. Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Technion City, Haifa, 32000, Israel

    Ilya Slavkin & Oleg V. Gendelman

  3. Department of Mechanical Engineering, University of Wisconsin at Madison, Madison, WI, 53706, USA

    Melih Eriten

  4. Department of Aerospace Engineering, University of Illinois at Urbana–Champaign, Urbana, IL, 61801, USA

    D. Michael McFarland & Lawrence A. Bergman

Authors
  1. Mehmet Kurt
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  2. Ilya Slavkin
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  3. Melih Eriten
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  4. D. Michael McFarland
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  5. Oleg V. Gendelman
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  6. Lawrence A. Bergman
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  7. Alexander F. Vakakis
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Correspondence to Mehmet Kurt.

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Kurt, M., Slavkin, I., Eriten, M. et al. Effect of 1:3 resonance on the steady-state dynamics of a forced strongly nonlinear oscillator with a linear light attachment. Arch Appl Mech 84, 1189–1203 (2014). https://doi.org/10.1007/s00419-014-0877-3

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  • DOI: https://doi.org/10.1007/s00419-014-0877-3

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