Abstract

Considerable attention has been given to nonlinear metamaterials because they offer some interesting phenomena such as solitons, frequency shifts, and tunable bandgaps. However, only little is known about the spectro-spatial properties of a wave propagating in nonlinear periodic chains, particularly, a cell with multiple nonlinear resonators. This problem is investigated here. Our study examines both hardening and softening nonlinearities in the chains and in the local resonators. Explicit expressions for the nonlinear dispersion relations are derived by the method of multiple scales. We validate our analytical results using numerical simulations. The numerical simulation is based on spectro-spatial analysis using signal processing techniques such as spatial-spectrogram and wave filtering. The spectro-spatial analysis provides detailed information about the interactions of dispersive and nonlinear phenomena of waveform in both short- and long-wavelength domains. Furthermore, we validate and demonstrate the theoretically obtained bandgaps, wave distortion, and birth of solitary waves through a computational study using finite element software, ansys. The findings, in both theoretical and computational analyses, suggest that nonlinear resonators can have more effect on the waveform than the nonlinear chains. This observation is valid in both short and long wavelength limits.

References

1.
Hussein
,
M. I.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2014
, “
Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Out4look
,”
ASME Appl. Mech. Rev.
,
66
(
4
), p.
040802
. 10.1115/1.4026911
2.
Bertoldi
,
K.
,
Vitelli
,
V.
,
Christensen
,
J.
, and
van Hecke
,
M.
,
2017
, “
Flexible Mechanical Metamaterials
,”
Nat. Rev. Mater.
,
2
(
11
), p.
17066
. 10.1038/natrevmats.2017.66
3.
Sigalas
,
M. M.
, and
Economou
,
E. N.
,
1992
, “
Elastic and Acoustic Wave Band Structure
,”
J. Sound Vib.
,
158
(
2
), pp.
377
382
. 10.1016/0022-460X(92)90059-7
4.
Sigalas
,
M.
, and
Economou
,
E. N.
,
1993
, “
Band Structure of Elastic Waves in Two Dimensional Systems
,”
Solid. State. Commun.
,
86
(
3
), pp.
141
143
. 10.1016/0038-1098(93)90888-T
5.
Kushwaha
,
M. S.
,
Halevi
,
P.
,
Dobrzynski
,
L.
, and
Djafari-Rouhani
,
B.
,
1993
, “
Acoustic Band Structure of Periodic Elastic Composites
,”
Phys. Rev. Lett.
,
71
(
13
), p.
2022
. 10.1103/PhysRevLett.71.2022
6.
Kushwaha
,
M. S.
,
Halevi
,
P.
,
Martinez
,
G.
,
Dobrzynski
,
L.
, and
Djafari-Rouhani
,
B.
,
1994
, “
Theory of Acoustic Band Structure of Periodic Elastic Composites
,”
Phys. Rev. B
,
49
(
4
), p.
2313
. 10.1103/PhysRevB.49.2313
7.
Vasseur
,
J.
,
Djafari-Rouhani
,
B.
,
Dobrzynski
,
L.
,
Kushwaha
,
M.
, and
Halevi
,
P.
,
1994
, “
Complete Acoustic Band Gaps in Periodic Fibre Reinforced Composite Materials: the Carbon/epoxy Composite and Some Metallic Systems
,”
J. Phys.: Condens. Matter.
,
6
(
42
), p.
8759
. 10.1088/0953-8984/6/42/008
8.
Kushwaha
,
M. S.
,
1996
, “
Classical Band Structure of Periodic Elastic Composites
,”
Int. J. Mod. Phys. B.
,
10
(
9
), pp.
977
1094
. 10.1142/S0217979296000398
9.
Liu
,
Z.
,
Zhang
,
X.
,
Mao
,
Y.
,
Zhu
,
Y.
,
Yang
,
Z.
,
Chan
,
C. T.
, and
Sheng
,
P.
,
2000
, “
Locally Resonant Sonic Materials
,”
Science
,
289
(
5485
), pp.
1734
1736
. 10.1126/science.289.5485.1734
10.
Liu
,
L.
, and
Hussein
,
M. I.
,
2012
, “
Wave Motion in Periodic Flexural Beams and Characterization of the Transition Between Bragg Scattering and Local Resonance
,”
ASME J. Appl. Mech.
,
79
(
1
), p.
011003
. 10.1115/1.4004592
11.
Huang
,
G.
, and
Sun
,
C.
,
2010
, “
Band Gaps in a Multiresonator Acoustic Metamaterial
,”
ASME J. Vib. Acoust.
,
132
(
3
), p.
031003
. 10.1115/1.4000784
12.
Zhu
,
R.
,
Liu
,
X.
,
Hu
,
G.
,
Sun
,
C.
, and
Huang
,
G.
,
2014
, “
A Chiral Elastic Metamaterial Beam for Broadband Vibration Suppression
,”
J. Sound. Vib.
,
333
(
10
), pp.
2759
2773
. 10.1016/j.jsv.2014.01.009
13.
Kivshar
,
Y. S.
, and
Flytzanis
,
N.
,
1992
, “
Gap Solitons in Diatomic Lattices
,”
Phys. Rev. A.
,
46
(
12
), p.
7972
. 10.1103/PhysRevA.46.7972
14.
Nadkarni
,
N.
,
Daraio
,
C.
, and
Kochmann
,
D. M.
,
2014
, “
Dynamics of Periodic Mechanical Structures Containing Bistable Elastic Elements: From Elastic to Solitary Wave Propagation
,”
Phys. Rev. E
,
90
(
2
), p.
023204
. 10.1103/PhysRevE.90.023204
15.
Liang
,
B.
,
Yuan
,
B.
, and
Cheng
,
J.-c.
,
2009
, “
Acoustic Diode: Rectification of Acoustic Energy Flux in One-Dimensional Systems
,”
Phys. Rev. Lett.
,
103
(
10
), p.
104301
. 10.1103/PhysRevLett.103.104301
16.
Manimala
,
J. M.
, and
Sun
,
C.
,
2016
, “
Numerical Investigation of Amplitude-Dependent Dynamic Response in Acoustic Metamaterials With Nonlinear Oscillators
,”
J. Acoust. Soc. Am.
,
139
(
6
), pp.
3365
3372
. 10.1121/1.4949543
17.
Nayfeh
,
A. H.
,
2011
,
Introduction to Perturbation Techniques
,
John Wiley & Sons
,
New York
.
18.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
2008
,
Nonlinear Oscillations
,
John Wiley & Sons
,
New York
.
19.
Narisetti
,
R. K.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2010
, “
A Perturbation Approach for Predicting Wave Propagation in One-Dimensional Nonlinear Periodic Structures
,”
ASME J. Vib. Acoust.
,
132
(
3
), p.
031001
. 10.1115/1.4000775
20.
Manktelow
,
K.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2011
, “
Multiple Scales Analysis of Wave-Wwave Interactions in a Cubically Nonlinear Monoatomic Chain
,”
Nonlin. Dyn.
,
63
(
1–2
), pp.
193
203
. 10.1007/s11071-010-9796-1
21.
Lazarov
,
B. S.
, and
Jensen
,
J. S.
,
2007
, “
Low-frequency Band Gaps in Chains With Attached Non-Linear Oscillators
,”
Int. J. Non-Linear Mech.
,
42
(
10
), pp.
1186
1193
. 10.1016/j.ijnonlinmec.2007.09.007
22.
Khajehtourian
,
R.
, and
Hussein
,
M. I.
,
2014
, “
Dispersion Characteristics of a Nonlinear Elastic Metamaterial
,”
Aip Adv.
,
4
(
12
), p.
124308
. 10.1063/1.4905051
23.
Hussein
,
M.
, and
Khajehtourian
,
R.
,
2018
, “
Nonlinear Bloch Waves and Balance Between Hardening and Softening Dispersion
,”
Proc. R. Soc. A: Math., Phys. Eng. Sci.
,
474
(
2217
), p.
20180173
.
24.
Fang
,
X.
,
Wen
,
J.
,
Benisty
,
H.
, and
Yu
,
D.
,
2020
, “
Ultrabroad Acoustical Limiting in Nonlinear Metamaterials Due to Adaptive-Broadening Band-Gap Effect
,”
Phys. Rev. B
,
101
(
10
), p.
104304
. 10.1103/PhysRevB.101.104304
25.
Xu
,
X.
,
Barnhart
,
M. V.
,
Fang
,
X.
,
Wen
,
J.
,
Chen
,
Y.
, and
Huang
,
G.
,
2019
, “
A Nonlinear Dissipative Elastic Metamaterial for Broadband Wave Mitigation
,”
Int. J. Mech. Sci.
,
164
, p.
105159
. 10.1016/j.ijmecsci.2019.105159
26.
Liang
,
B.
,
Guo
,
X.
,
Tu
,
J.
,
Zhang
,
D.
, and
Cheng
,
J.
,
2010
, “
An Acoustic Rectifier
,”
Nat. Mater.
,
9
(
12
), p.
989
. 10.1038/nmat2881
27.
Li
,
X.-F.
,
Ni
,
X.
,
Feng
,
L.
,
Lu
,
M.-H.
,
He
,
C.
, and
Chen
,
Y.-F.
,
2011
, “
Tunable Unidirectional Sound Propagation Through a Sonic-Crystal-Based Acoustic Diode
,”
Phys. Rev. Lett.
,
106
(
8
), p.
084301
. 10.1103/PhysRevLett.106.084301
28.
Boechler
,
N.
,
Theocharis
,
G.
, and
Daraio
,
C.
,
2011
, “
Bifurcation-Based Acoustic Switching and Rectification
,”
Nat. Mater.
,
10
(
9
), p.
665
. 10.1038/nmat3072
29.
Moore
,
K. J.
,
Bunyan
,
J.
,
Tawfick
,
S.
,
Gendelman
,
O. V.
,
Li
,
S.
,
Leamy
,
M.
, and
Vakakis
,
A. F.
,
2018
, “
Nonreciprocity in the Dynamics of Coupled Oscillators With Nonlinearity, Asymmetry, and Scale Hierarchy
,”
Phys. Rev. E
,
97
(
1
), p.
012219
. 10.1103/PhysRevE.97.012219
30.
Ma
,
C.
,
Parker
,
R. G.
, and
Yellen
,
B. B.
,
2013
, “
Optimization of An Acoustic Rectifier for Uni-directional Wave Propagation in Periodic Mass–Spring Lattices
,”
J. Sound. Vib.
,
332
(
20
), pp.
4876
4894
. 10.1016/j.jsv.2013.04.013
31.
Ganesh
,
R.
, and
Gonella
,
S.
,
2013
, “
Spectro-Spatial Wave Features As Detectors and Classifiers of Nonlinearity in Periodic Chains
,”
Wave Motion
,
50
(
4
), pp.
821
835
. 10.1016/j.wavemoti.2013.05.002
32.
Zhou
,
W.
,
Li
,
X.
,
Wang
,
Y.
,
Chen
,
W.
, and
Huang
,
G.
,
2018
, “
Spectro-Spatial Analysis of Wave Packet Propagation in Nonlinear Acoustic Metamaterials
,”
J. Sound. Vib.
,
413
, pp.
250
269
. 10.1016/j.jsv.2017.10.023
33.
Bukhari
,
M.
, and
Barry
,
O.
,
2019
, “
Spectro-Spatial Analyses of a Nonlinear Metamaterial With Multiple Nonlinear Local Resonators
,”
Nonlinear Dyn.
,
99
(
2
), pp.
1
22
.
34.
Bukhari
,
M. A.
, and
Barry
,
O. R.
,
2019
, “
On the Spectro-Spatial Wave Features in Nonlinear Metamaterials with Multiple Local Resonators
,”
ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Anaheim, CA
.
35.
Bloch
,
F.
,
1929
, “
Über Die Quantenmechanik Der Elektronen in Kristallgittern
,”
Zeitschrift für physik
,
52
(
7–8
), pp.
555
600
. 10.1007/BF01339455
36.
Floquet
,
G.
,
1883
, “
Sur Les équations Différentielles Linéaires à Coefficients Périodiques
,”
Annates Scientifiques de VEcole Normale superie
,
12
, pp.
47
88
.
37.
Bukhari
,
M. A.
, and
Barry
,
O. R.
,
2019
, “
Nonlinear Metamaterials With Multiple Local Mechanical Resonators: Analytical and Numerical Analyses
,”
NODYCON 2019 The First International Nonlinear Dynamics Conference
,
Rome
.
38.
Manimala
,
J. M.
,
2014
,
Dynamic behavior of acoustic metamaterials and metaconfigured structures with local oscillators
. Ph.D. thesis,
Purdue University
.
39.
Hu
,
G.
,
Tang
,
L.
, and
Das
,
R.
,
2018
, “
Internally Coupled Metamaterial Beam for Simultaneous Vibration Suppression and Low Frequency Energy Harvesting
,”
J. Appl. Phys.
,
123
(
5
), p.
055107
. 10.1063/1.5011999
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