On control of harmonic waves in an acoustic metamaterial

doi:10.1016/j.mechrescom.2021.103745

Mechanics Research Communications

Volume 116, September 2021, 103745

https://doi.org/10.1016/j.mechrescom.2021.103745 Get rights and content

Highlights

•
The boundary excitation of waves in the metamaterial mass-in-mass lattice model.
•
Formation of periodic waves close to the exact solution.
•
Development of a control allowing waves propagation inside a band gap.

Abstract

Generation of harmonic waves in a mass-in mass model of an acoustic metamaterial is considered. It is shown numerically that outside the band gap, the boundary harmonic excitation gives rise to a formation of periodic waves whose profile gradually coincides with the profile of the exact traveling wave harmonic solution. No waves are generated by the boundary excitation inside the band gap. A switch-on/off control is developed to achieve the formation of the harmonic waves inside the band gap.

Introduction

The interest to the study of the acoustic metamaterials has grown considerably in the recent years, see [1], [2], [3], [4], [5], [6], [7], [8], [9] and references therein. The linear acoustic metamaterials draw more attention [2], [3], [5]. One of the simplest but instructive model is the mas-in-mass model. It allows to describe the band gap in the dispersion relation [2], [3], negative effective mass and other features related to a metamaterial.

Experimental realization of the mass-in-mass model may be found in [2], [10], [11], [12]. The existence of a band gap was confirmed there. However, no periodic waves were recorded. The metamaterial has been constructed in [10] so as to change the internal oscillator by an electric signal. Similarly the external electromagnetic signals were used in [13], [14] to develop the active tunable acoustic metamaterials. One can note the luck of theoretical results concerning generation of the harmonic waves and their control.

In this paper we study numerically generation of harmonic waves in a linear metamaterial mass-in-mass model. The boundary harmonic excitation is shown to produce both the acoustic and optic harmonic waves outside the band gap while no wave propagation is obtained inside the band gap. Then the switch-on/off control is developed to see how to change the shape of the harmonic waves inside and outside the band gap.

Section snippets

Statement of problem

Consider a chain when interaction between the masses, $m$ , is modeled by linearly-elastic springs [3]. An elastic internal oscillator is modeled by the additional masses $m_{1}$ , attached by springs to each mass $m$ in the chain, and this interaction is also linear and elastic. Masses $m_{1}$ do not interact directly between themselves. The displacement of the mass $m$ with the number $n$ is denoted by $x_{n}$ , while that of $m_{1}$ is denoted by $y_{n}$ . $\ddot{x_{n}} = β_{0} (x_{n - 1} - 2 x_{n} + x_{n + 1}) + η β_{1} (y_{n} - x_{n}),$ $\ddot{y_{n}} = - β_{1} (y_{n} - x_{n}) .$ Here $η = m_{1} ∕ m$ , while the

Generation of harmonic waves

The analysis in the previous Section is based on the particular traveling wave solution which requires specific initial and boundary conditions for its realization. However, experimental generation of the waves in a metamaterial uses the boundary excitation of the waves. In particular, the following initial and boundary conditions for $u$ can be employed, $u (0, t) = \frac{β_{1} - ω^{2}}{β_{1}} B sin (ω t), u (x, 0) = 0, u {(x, 0)}_{t} = 0,$ while for $v$ we assume that $v (0, t) = B sin (ω t), v (x, 0) = 0, v {(x, 0)}_{t} = 0,$

This problem can be solved numerically.

Control of harmonic waves

The use of electromagnetic signals in [10], [13], [14] to change the internal properties of a metamaterial suggests the mechanism of a control based on an instant switch-on. It is modeled by the unit step function $H (t_{0} - t)$ that switches-off the coupling in Eqs. (3), (4) at $t = t_{0}$ . Thus, for $u$ we have $u_{t t} = β_{0} h^{2} u_{x x} + η β_{1} (v - u) H (t_{0} - t)$ Also, the unit step function should be added in equation of motion (4) and boundary condition for $v$ (10). The switch-on time for the control, $t_{0}$ , is chosen equal to $t_{N} ∕ 4$ ,

Discussion

The boundary excitation can generate harmonic waves in the metamaterial in an agreement with the results obtained on the basis of the particular traveling wave solution. However, the area of the values of the frequency where no periodic waves propagates turns out wider than that predicted by the analysis of the dispersion relation. Besides the band gap zone where no waves propagate, there is an interval where non-periodic waves travel.

The switch-on control allow us to provide harmonic waves

Acknowledgment

This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Project No. 075-15-2021-573).

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View full text

On control of harmonic waves in an acoustic metamaterial

Highlights

Abstract

Introduction

Section snippets

Statement of problem

Generation of harmonic waves

Control of harmonic waves

Discussion

Acknowledgment

Internat. J. Engrg. Sci.

Mech. Res. Commun.

Mech. Res. Commun.

Theor. Appl. Mech. Lett.

IFAC Proc. Vol.

Controlling sound with acoustic metamaterials

Nat. Rev. Mater.

Acoustic metamaterials: From local resonances to broad horizons

Sci. Adv.