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Research Papers

Metamaterial With Local Resonators Coupled by Negative Stiffness Springs for Enhanced Vibration Suppression

[+] Author and Article Information
Guobiao Hu

Department of Mechanical Engineering,
University of Auckland,
20 Symonds Street,
Auckland 1010, New Zealand
e-mail: ghu211@aucklanduni.ac.nz

Lihua Tang

Department of Mechanical Engineering,
University of Auckland,
20 Symonds Street,
Auckland 1010, New Zealand
e-mail: l.tang@auckland.ac.nz

Jiawen Xu

Jiangsu Key Lab of Remote Measurement and Control,
School of Instrument Science and Engineering,
Southeast University,
Nanjing, Jiangsu 210096, China
e-mail: jiawen.xu@seu.edu.cn

Chunbo Lan

College of Aerospace Engineering,
Nanjing University of Aeronautics and Astronautics,
Nanjing, Jiangsu 210016, China
e-mail: chunbolan@nuaa.edu.cn

Raj Das

School of Engineering,
RMIT University,
GPO Box 2476,
Melbourne, VIC 3001, Australia
e-mail: raj.das@rmit.edu.au

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received March 18, 2019; final manuscript received May 20, 2019; published online June 4, 2019. Assoc. Editor: Haleh Ardebili.

J. Appl. Mech 86(8), 081009 (Jun 04, 2019) (14 pages) Paper No: JAM-19-1120; doi: 10.1115/1.4043827 History: Received March 18, 2019; Accepted May 20, 2019

In recent years, metamaterials for the applications in low-frequency vibration suppression and noise reduction have attracted numerous research interests. This paper proposes a metamaterial system with local resonators from adjunct unit cells coupled by negative stiffness springs. Frist, a lumped parameter model of the system is developed, and a stability criterion is derived. The band structure of the infinite lattice model is calculated. The result reveals the appearance of extra band gaps in the proposed metamaterial. A parametric study shows that the first extra band gap can be tuned to ultralow frequency by controlling the negative stiffness of the coupling springs. A transmittance analysis of the finite lattice model verifies the predictions obtained from the band structure analysis. Subsequently, the work is extended to a distributed parameter metamaterial beam model with the proposed configuration of coupled local resonators. The stability analysis shows that the infinitely long metamaterial beam becomes unstable as long as the stiffness of the coupling spring becomes negative. For the finitely long metamaterial beam, the stability could be achieved for negative coupling springs of given stiffnesses. The effects of the number of cells and the lattice constant on the system stability are investigated. The transmittance of the finitely long metamaterial beam is calculated. The result shows that due to the restriction on the tunability of negative stiffness for the proposed metamaterial beam, a quasistatic vibration suppression region can only be achieved when the number of cells is small.

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Figures

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Fig. 1

(a) Infinite lattice model and (b) finite lattice model of the metamaterial with local resonators coupled by negative stiffness springs

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Fig. 2

Critical βc versus μ and α

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Fig. 3

Dimensionless effective mass meff/m1 versus dimensionless frequency Ω

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Fig. 4

Force and displacement interactions at the interfaces

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Fig. 5

Real part of complex band structures of the proposed metamaterial with different β: (a) β = 0 (i.e., the conventional metamaterial), (b) β = −0.2, (c) β = −0.4, and (d) β = −0.439

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Fig. 6

Imaginary part of complex band structures of the proposed metamaterial with different values of β: (a) β = 0 (i.e., the conventional metamaterial), (b) β = −0.2, (c) β = −0.4, and (d) β = −0.439

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Fig. 7

Evolution of band gaps of the proposed metamaterial with varying β

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Fig. 8

Evolution of the band gaps of the proposed metamaterials at critical βc with varying μ and α

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Fig. 9

Transmittances of the finite lattice model of the proposed metamaterial consisting of (a) five cells with different values of β and (b) different number of cells at critical βc

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Fig. 10

(a) Infinitely long model and (b) finitely long model of the proposed metamaterial beam with local resonators coupled by negative stiffness springs. The host beam is discretised as finite elements.

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Fig. 11

Convergence plots of the first four natural frequencies with refinement of mesh: (a) first natural frequency, (b) second natural frequency, (c) third natural frequency, and (d) fourth natural frequency

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Fig. 12

Variation of βc (a) with an increase in the number of cells (the lattice constant is fixed at 50 mm) and (b) with an increase in lattice constant (the number of cells is fixed at 6)

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Fig. 13

Transmittances of the proposed metamaterial beam with different β and different number of cells: (a) 3 cells, (b) 5 cells, (c) 10 cells, and (d) 15 cells

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Fig. 14

Equivalent electrical and mechanical representations of a piezoelectric patch shunted to a negative capacitance

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