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Ultra-broad bandgap induced by hybrid hardening and softening nonlinearity in metastructure

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Abstract

Metamaterials are artificial microstructured media that exhibit unique effective material properties and can be tailored to achieve negative properties that inhibit the propagation of acoustic or elastic waves. However, the effectiveness of linear metamaterials (LMs) based on resonance mechanisms is limited to a narrow frequency band. To overcome this limitation, nonlinear metamaterials (NLMs) have been investigated for enlarged bandwidth and wave phenomena beyond the linear systems. This paper presents a new mechanism for achieving an ultra-broad bandgap using a chain of triplets of resonators, based on the combination of hardening-plus-softening nonlinearity. The paper begins by investigating the bandgap and transmission characteristics of NLMs with different arrangements of single-hardening or single-softening nonlinear spring element. Explicit expressions for the nonlinear dispersion relations have been derived through a perturbation method. The paper then explores the combination of hardening-plus-softening nonlinearity to achieve an ultra-broad bandgap, which is more than twice as wide as the bandgap in the corresponding LM. The transmission characteristics are investigated analytically and validated numerically, providing evidence for the existence and accuracy of the predicted ultra-broad bandgap. The paper also examines nonlinear phenomena such as the dual wavevector and inflection point of transmissions in detail. Finally, the physical origin of the ultra-broad bandgap is elaborated through the nonlinear frequency response of a unit cell solved using a high-order harmonic balance method. The findings of this study are expected to provide a new strategy for broadening vibration suppression and offer new insight into the behavior of nonlinear periodic structures.

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Abbreviations

\({A}_{1}^{0}\) :

Amplitude of incident wave

\({A}_{b}\) :

Excitation amplitude applied at the first unit cell in numerical analysis

\(a\) :

Lattice constant

\({c}_{1}\) :

Secular terms

c.c :

Complex conjugate

\(i\) :

Imaginary unit \(\sqrt{-1}\)

\({k}_{1}\), \({k}_{2}\),\({k}_{3}\) :

Linear spring coefficient of unit cells from the external to the internal

LM:

Linear metamaterial

\({m}_{1}\), \({m}_{2}\),\({m}_{3}\) :

Discrete mass of unit cells from the external to the internal

n :

Location number of unit cell

NLM:

Nonlinear metamaterial

\(q\) :

The wave number

\({T}_{l}\) :

Transmission for linear system

\({T}_{nl}\) :

Transmission for nonlinear system

\({u}_{1}^{n}\), \({u}_{2}^{n}\),\({u}_{3}^{n}\) :

Displacement of \({m}_{1}\),\({m}_{2}\), \({m}_{3}\) For the nth unit cells

\({{u}_{j}^{n}}^{\left(0\right)}\) :

The zero-order term of the perturbation expansion of the displacement

\({{u}_{j}^{n}}^{\left(1\right)}\) :

The first-order term of the perturbation expansion of the displacement

\({x}_{i}\) :

The deformation of the corresponding spring

\({\gamma }_{2}\),\({\gamma }_{3}\) :

Non-dimensional nonlinear stiffness

\({\delta }_{2}\),\({\delta }_{3}\) :

Non-dimensional stiffness

\(\varepsilon \) :

Perturbation parameter

\({\theta }_{2}\),\({\theta }_{3}\) :

Non-dimensional mass

\(\kappa \) :

Dimensionless wave number

\(\lambda \) :

Coefficient of

\(\tau \) :

Non-dimensional time

\({\omega }_{0}\) :

Natural frequency

\({\Gamma }_{2}\),\({\Gamma }_{3}\) :

Nonlinear spring coefficients

\(\Omega \) :

Non-dimensional frequency

\({\Omega }_{0}\) :

Zero-order term of the perturbation expansion of the non-dimensional frequency

\({\Omega }_{1}\) :

First-order term of the perturbation expansion of the non-dimensional frequency

\({\Omega }_{L}^{1}\) :

Lower edge frequency of the first bandgap

\({\Omega }_{U}^{1}\) :

Upper edge frequency of the first bandgap

\({\Omega }_{L}^{2}\) :

Lower edge frequency of second bandgap

\({\Omega }_{U}^{2}\) :

Upper edge frequency of second bandgap

\({\Omega }_{c}\) :

Cut-off frequency

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Acknowledgements

This work was supported in part by the National Key Research and Development Program of China under Grant No. 2021YFE0110900 and in part by the National Natural Science Foundation of China (NSFC) under Grants No. U22B2078, 11991033 and 12202052. The involvement of Ivana Kovacic was supported by the Ministry of Science, Innovation and Technological Development of the Republic of Serbia via the NOLIMAST project.

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Authors and Affiliations

  1. School of Aerospace Engineering, Bei**g Institute of Technology, Bei**g, 100081, China

    Jianlei Zhao, Hao Zhou, Kaijun Yi & Rui Zhu

  2. Faculty of Technical Science, Centre of Excellence for Vibro-Acoustic Systems and Signal Processing, University of Novi Sad, 21000, Novi Sad, Serbia

    Ivana Kovacic

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Zhao, J., Zhou, H., Yi, K. et al. Ultra-broad bandgap induced by hybrid hardening and softening nonlinearity in metastructure. Nonlinear Dyn 111, 17687–17707 (2023). https://doi.org/10.1007/s11071-023-08808-w

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