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Applied Mathematics and Mechanics (English Edition) ›› 2017, Vol. 38 ›› Issue (1): 1-14.doi: https://doi.org/10.1007/s10483-016-2152-6

• 论文 •    下一篇

Primary resonance of traveling viscoelastic beam under internal resonance

Hu DING1, Linglu HUANG1, Xiaoye MAO1, Liqun CHEN1,2   

  1. 1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China;
    2. Department of Mechanics, College of Science, Shanghai University, Shanghai 200444, China
  • 收稿日期:2016-05-20 修回日期:2016-05-30 出版日期:2017-01-01 发布日期:2017-01-01
  • 通讯作者: Hu DING,E-mail:dinghu3@shu.edu.cn E-mail:dinghu3@shu.edu.cn
  • 基金资助:

    Project supported by the State Key Program of the National Natural Science Foundation of China (No. 11232009) and the National Natural Science Foundation of China (Nos. 11372171 and 11422214)

Primary resonance of traveling viscoelastic beam under internal resonance

Hu DING1, Linglu HUANG1, Xiaoye MAO1, Liqun CHEN1,2   

  1. 1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China;
    2. Department of Mechanics, College of Science, Shanghai University, Shanghai 200444, China
  • Received:2016-05-20 Revised:2016-05-30 Online:2017-01-01 Published:2017-01-01
  • Contact: Hu DING E-mail:dinghu3@shu.edu.cn
  • Supported by:

    Project supported by the State Key Program of the National Natural Science Foundation of China (No. 11232009) and the National Natural Science Foundation of China (Nos. 11372171 and 11422214)

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摘要：

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.

关键词: internal resonance, primary resonance, nonlinear vibration, viscoelasticity, traveling beam

Abstract:

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: primary resonance, viscoelasticity, traveling beam, nonlinear vibration, internal resonance

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