Nonlinear Dynamics

, Volume 95, Issue 3, pp 2367–2382 | Cite as

Nonlinear vibration of a slightly curved beam with quasi-zero-stiffness isolators

  • Hu Ding
  • Li-Qun ChenEmail author
Original Paper


Bending vibration of isolated structures has always been neglected when the vibration isolation was studied. Isolated structures have usually been treated as discrete systems. In this study, dynamics of a slightly curved beam supported by quasi-zero-stiffness systems are firstly presented. In order to achieve quasi-zero-stiffness, a nonlinear isolation system is implemented via three linear springs. A nonlinear dynamic model of the slightly curved beam with nonlinear isolations is established. It includes square nonlinearity, cubic nonlinearity, and nonlinear boundaries. Then, the mode functions and the frequencies of the curved beam with elastic boundaries are derived. The schemes of the finite difference method (FDM) and the Galerkin truncation method (GTM) are, respectively, proposed to obtain nonlinear responses of the curved beam with nonlinear boundaries. Numerical results demonstrate that both the GTM and the FDM yield accurate solutions for the nonlinear dynamics of curved structures with nonsimple boundaries. The multi-mode resonance characteristics of the curved beam affect the vibration isolation efficiency. The quasi-zero-stiffness isolators reduce the transmissibility of modal resonances and provide a promising future for isolating the bending vibration of the flexible structure. However, the initial curvature significantly increases the resonant frequency of the flexible structure, and thus the frequency range of the effective vibration isolation is narrower. Furthermore, the quadratic nonlinear terms in the curved beam make the dynamic phenomenon more complicated. Therefore, it is more challenging and necessary to investigate the isolation of the bending vibration of the initial curved structure.


Curved beam Nonlinear vibration Quasi-zero-stiffness Nonlinear isolation 



The authors would wish to thank anonymous reviewers for those critical comments. The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Nos. 11772181, 11832009, and 11572182), the Program of Shanghai Municipal Education Commission (No. 17SG38), and the Key Research Projects of Shanghai Science and Technology Commission (No. 18010500100).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiChina
  2. 2.Harbin Institute of TechnologyShenzhenChina

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