Applied Mathematics and Mechanics

, Volume 38, Issue 1, pp 1–14 | Cite as

Primary resonance of traveling viscoelastic beam under internal resonance

  • Hu DingEmail author
  • Linglu Huang
  • Xiaoye Mao
  • Liqun Chen
Open Access


Under the 3:1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The undetermined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.

Key words

traveling beam nonlinear vibration viscoelasticity primary resonance internal resonance 

Chinese Library Classification


2010 Mathematics Subject Classification



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Copyright information

© The Author(s) 2016

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

This article is published with open access at, corrected publication 03/2018

The original article has been corrected.

Authors and Affiliations

  • Hu Ding
    • 1
    Email author
  • Linglu Huang
    • 1
  • Xiaoye Mao
    • 1
  • Liqun Chen
    • 1
    • 2
  1. 1.Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy EngineeringShanghai UniversityShanghaiChina
  2. 2.Department of Mechanics, College of ScienceShanghai UniversityShanghaiChina

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