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Nonlinear Dynamics

, Volume 95, Issue 3, pp 2543–2554 | Cite as

Nonlinear vibration of a beam with asymmetric elastic supports

  • Hu DingEmail author
  • Yi Li
  • Li-Qun Chen
Original Paper

Abstract

Under the conditions of horizontal placement and only considering geometric nonlinearity, depending on the boundary constraints, primary resonances of an elastic beam exhibit either hardening or softening nonlinear behavior. In this paper, the conversion of softening nonlinear characteristics to hardening characteristics is studied by using the multi-scale perturbation method. Therefore, in a local sense, the condition is established for the resonance of the elastic beam exhibits only linear characteristics by finding the balance between asymmetric elastic support and geometric nonlinearity. A viscoelastic beam supported by vertical springs is proposed with nonrotatable left boundary and freely rotatable right end. In order to truncate the continuous system, natural frequencies and modes of the proposed asymmetric beam are analyzed. The steady-state responses of the beam excited by a distributed harmonic force are, respectively, obtained by an approximate analytical method and a numerical approach. Under the condition that the beam is placed horizontally, the transition from the cantilever state to the clamped–pinned state is demonstrated by constructing different asymmetry support conditions. The resonance peak of the first-order primary resonance is used to demonstrate the transition from softening nonlinear characteristics to the hardening characteristics. This research shows that the transformation from softening characteristics to hardening characteristics caused by asymmetric elastic support and geometric nonlinearity exists only in the first-order mode resonance.

Keywords

Geometric nonlinearity Elastic beam Asymmetric elastic boundaries Nonlinear characteristics 

Notes

Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China [Grant Numbers 11772181, 11422214], the “Dawn” Program of Shanghai Education Commission (Grant Number 17SG38) and the Innovation Program of Shanghai Municipal Education Commission [Grant Number 2017-01-07-00-09-E00019].

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interests.

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiChina
  2. 2.Shanghai Key Laboratory of Mechanics in Energy EngineeringShanghaiChina
  3. 3.Department of MechanicsShanghai UniversityShanghaiChina

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