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Research Papers

Elastic Metamaterials With Low-Frequency Passbands Based on Lattice System With On-Site Potential

[+] Author and Article Information
Yongquan Liu

Department of Mechanics and
Engineering Science,
College of Engineering,
Peking University,
Beijing 100871, China;
School of Aeronautics and Astronautics,
Purdue University,
West Lafayette, IN 47907

Xiaohui Shen

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

Xianyue Su

Department of Mechanics and
Engineering Science,
College of Engineering,
Peking University,
Beijing 100871, China

C. T. Sun

School of Aeronautics and Astronautics,
Purdue University,
West Lafayette, IN 47907
e-mail: sun@purdue.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 August 20, 2015; final manuscript received December 12, 2015; published online January 21, 2016. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 138(2), 021011 (Jan 21, 2016) (10 pages) Paper No: VIB-15-1335; doi: 10.1115/1.4032326 History: Received August 20, 2015; Revised December 12, 2015

An elastic metamaterial with a low-frequency passband is proposed by imitating a lattice system with linear on-site potential. It is shown that waves can only propagate in the tunable passband. Then, two kinds of elastic metamaterials with double passbands are designed. Great wave attenuation performance can be obtained at frequencies between the two passbands for locally resonant type metamaterials, and at both low and high frequencies for the diatomic type metamaterials. Finally, the strategy to design two-dimensional (2D) metamaterials is demonstrated. The present method can be used to design new types of small-size waveguides, filters, and other devices for elastic waves.

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Figures

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Fig. 1

The model of monoatomic lattice system with linear on-site potential

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Fig. 2

(a) The dispersion curve and (b) effective parameters of monoatomic lattice system with linear on-site potential (m1=1, k1=2, and k2=1)

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Fig. 3

N-period monoatomic lattice system with effective mass

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Fig. 4

Analytically calculated (curves) and numerically simulated (dots) transmittance of lattice systems with 5, 10, and 15 periods, respectively

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Fig. 5

The continuum model of elastic metamaterials with one low-frequency passband

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Fig. 6

(a) Transmission coefficients of a 9-period system structure with and without steel resonator and (b) transmittance of a 9-period structure for different resonators

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Fig. 7

The locally resonant model of elastic metamaterials with two low-frequency passbands

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Fig. 8

(a) The dispersion curve, (b) the attenuation factor, and (c) effective parameters of locally resonant type elastic metamaterials with two passbands (m1=1, m2=2, k1=2, k2=1, and k3=3)

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Fig. 9

Transmittance of lattice systems (a) versus periods of the systems for m2=2 and (b) versus θ=m2/m1 for a 10-period system. All results are for m1=1, k1=2, k2=1, and k3=3.

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Fig. 10

Transmission coefficients of a 9-period system for different radii of the outer resonator

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Fig. 11

The diatomic model of elastic metamaterials with two passbands

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Fig. 12

(a) The dispersion curve, (b) the attenuation factor, and (c) effective parameters of diatomic type elastic metamaterials with two passbands (m1=1, m2=2, k1=2, k2=1, and k3=3)

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Fig. 13

N-period diatomic lattice system with effective mass. (a) The output mass is different from the input one and (b) the output mass is the same as the input one.

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Fig. 14

Transmittance of lattice systems with different numbers of unit cells for (a) case 1 and (b) case 2. All results are for m1=1, k1=2, k2=1, and k3=3.

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Fig. 15

Transmission coefficients of the diatomic type continuum metamaterials with two passbands. The solid line and the line with circles represent cases 1 and 2, respectively.

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Fig. 16

(a) The 2D counterpart of monoatomic lattice system with linear on-site potential, (b) the continuum model, (c) the dispersion curve of the continuum model, with maps to show displacement fields of the selected points, and (d) variation of the passband with respect to the side length of the resonator C

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Fig. 17

Transmission of the 2D continuum model shown in Fig. 16(b) through a finite sample. (a) The numerical setup for transmission computations along the ΓM direction, (b) transmission along the ΓM direction for P wave and S wave input excitations, and (c) transmission along the ΓX direction for P wave and S wave input excitations.

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Fig. 18

(a) The 2D counterpart of locally resonant system with linear on-site potential, (b) the continuum model, (c) the dispersion curve of the continuum model, (d) the first, second, fifth, and sixth Bloch modes at point X of the Brillouin zone, and (e) variation of both passbands with respect to the side length of the outer resonator C1

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Fig. 19

(a) The 2D counterpart of diatomic lattice system with linear on-site potential, (b) the continuum model, (c) the dispersion curve of the continuum model, (d) the first, second, sixth, and seventh Bloch modes at point X of the Brillouin zone, and (e) variation of both passbands with respect to the side length of the central resonator C1

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Fig. 20

Unit cell of the monoatomic lattice system and the way to calculate its effective stiffness

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Fig. 21

The way to calculate (a) the effective stiffness and (b) the effective mass by analyzing the unit cell of diatomic lattice systems

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