Archive of Applied Mechanics

, Volume 84, Issue 8, pp 1189–1203 | Cite as

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

  • Mehmet KurtEmail author
  • Ilya Slavkin
  • Melih Eriten
  • D. Michael McFarland
  • Oleg V. Gendelman
  • Lawrence A. Bergman
  • Alexander F. Vakakis


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.


Frequency–energy plots 1:3 resonance Nonlinear steady-state solutions Nonlinear targeted energy transfer 



Degree of freedom


Frequency–energy plot


Nonlinear normal mode


Complexification-averaging method


Symmetric 1:1 resonance branch


Symmetric 1:3 resonance branch


Targeted energy transfer


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Mehmet Kurt
    • 1
    Email author
  • Ilya Slavkin
    • 2
  • Melih Eriten
    • 3
  • D. Michael McFarland
    • 4
  • Oleg V. Gendelman
    • 2
  • Lawrence A. Bergman
    • 4
  • Alexander F. Vakakis
    • 1
  1. 1.Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana–ChampaignUrbanaUSA
  2. 2.Faculty of Mechanical EngineeringTechnion—Israel Institute of TechnologyTechnion CityIsrael
  3. 3.Department of Mechanical EngineeringUniversity of Wisconsin at MadisonMadisonUSA
  4. 4.Department of Aerospace EngineeringUniversity of Illinois at Urbana–ChampaignUrbanaUSA

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