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Research Papers

Analysis of Dynamic Behavior of the Finite Elastic Metamaterial-Based Structure With Frequency-Dependent Properties

[+] Author and Article Information
X. H. Shen

School of Aeronautics and Astronautics,
Purdue University,
West Lafayette, IN 47907;
Department of Mechanical and
Aerospace Engineering,
University of Missouri,
Columbia, MO 65211

C. T. Sun

School of Aeronautics and Astronautics,
Purdue University,
West Lafayette, IN 47907

M. V. Barnhart

Department of Mechanical and
Aerospace Engineering,
University of Missouri,
Columbia, MO 65211

G. L. Huang

Department of Mechanical and
Aerospace Engineering,
University of Missouri,
Columbia, MO 65211
e-mail: huangg@missouri.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 27, 2017; final manuscript received January 4, 2018; published online February 9, 2018. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 140(3), 031012 (Feb 09, 2018) (11 pages) Paper No: VIB-17-1284; doi: 10.1115/1.4038950 History: Received June 27, 2017; Revised January 04, 2018

For practical applications of the elastic metamaterials, dynamic behavior of finite structures made of elastic metamaterials with frequency dependent properties are analyzed theoretically and numerically. First, based on a frequency-dependent mass density and Young's modulus of the effective continuum, the global dynamic response of a finite rod made of elastic metamaterials is studied. It is found that due to the variation of the effective density and Young's modulus, the natural frequency distribution of the finite structure is altered. Furthermore, based on the spectral approach, the general wave amplitude transfer function is derived before the final transmitted wave amplitude for the finite-layered metamaterial structure with decreasing density is obtained using the mathematical induction method. The analytical analysis and finite element solutions indicate that the increased transmission wave displacement amplitude and reduced stress amplitude can be controlled by the impedance mismatch of the adjacent layers of the layered structure.

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Figures

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Fig. 1

Infinite mass-in-mass lattice model

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Fig. 2

Infinite effective mass lattice model

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Fig. 3

Unit cell of monatomic effective mass lattice model (left) and effective continuum representation (right)

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Fig. 4

(a) Dispersion curves of lattice system and (b) effective mass and Young's modulus parameters

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Fig. 5

Finite mass-in-mass lattice model and analogous effective continuum rod (free-free boundary condition)

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Fig. 6

Finite mass-in-mass lattice model and effective continuum rod (fixed-free end)

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Fig. 7

Comparison of (a) dispersion curves with band gap and (b) natural frequencies of metamaterial rod

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Fig. 8

Free–free boundary condition of finite mass-in-mass lattice model in abaqus

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Fig. 9

Natural frequencies of the elastic metamaterial rod obtained from analytical (Eq.(18)) and numerical models with 100 resonators and 20 resonators

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Fig. 10

Finite lattice model and effective continuum rod with displacement input

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Fig. 11

Displacement comparison of the remote end in the effective continuum rod and metamaterial-based rod for (a) natural frequency distribution comparison and (b) natural frequency concentration in the first passing band

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Fig. 12

Diagram of layered structure

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Fig. 13

(a) Three section mass-in-mass lattice model with unit displacement input and (b) three section effective continuum model with unit displacement input

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Fig. 14

Transmitted wave time history (a) effective continuum model and (b) mass-in-mass lattice model

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Fig. 15

(a) Three section effective continuum model with unit force input and (b) time history comparison for input stress (dashed line) and transmitted stress (solid line)

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Fig. 16

Five section effective continuum model with unit displacement input

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Fig. 17

Transmitted wave time history for the (a) maximum design and (b) minimum design

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