Wave dispersion in a string carrying periodically distributed masses is investigated analytically and experimentally. The effect of the string's geometric nonlinearity on its wave propagation characteristics is analyzed through a lumped parameter model yielding coupled Duffing oscillators. Dispersion frequency shifts are predicted that correspond to the hardening behavior of the nonlinear chain and that relate well to the backbone of individual Duffing oscillators. Experiments conducted on a string of finite length illustrate the relation between measured resonances and the dispersion properties of the medium. Specifically, the locus of resonance peaks in the frequency/wavenumber domain outlines the dispersion curve and highlights the existence of a frequency bandgap. Moreover, amplitude-dependent resonance shifts induced by the string nonlinearity confirm the hardening characteristics of the dispersion curve. Analytical and experimental results provide a critical link between nonlinear dispersion frequency shifts and the backbone curves intrinsic to nonlinear frequency response functions. Moreover, the study confirms that amplitude-dependent wave properties for nonlinear periodic systems may be exploited for tunability of wave transport characteristics such as frequency bandgaps and wave speeds.

References

References
1.
Kittel
,
C.
,
1976
,
Introduction to Solid State Physics
,
5th ed.
,
Wiley
,
New York
.
2.
Olsson
,
R. H.
, and
El-Kady
,
I.
,
2009
, “
Microfabricated Phononic Crystal Devices and Applications
,”
Meas. Sci. Tech.
,
20
(
1
), p.
012002
.10.1088/0957-0233/20/1/012002
3.
Elachi
,
C.
,
1976
, “
Waves in Active and Passive Periodic Structures: A Review
,”
Proc. IEEE
,
64
(
12
), pp.
1666
1698
.10.1109/PROC.1976.10409
4.
Ruzzene
,
M.
,
Scarpa
,
F.
, and
Soranna
,
F.
,
2003
, “
Wave Beaming Effects in Two-Dimensional Cellular Structures
,”
Smart Mat. Struct.
,
12
(
3
), pp.
363
372
.10.1088/0964-1726/12/3/307
5.
Khelif
,
A.
,
Djafari-Rouhani
,
B.
,
Vasseur
,
J.
, and
Deymier
,
P.
,
2003
, “
Transmission and Dispersion Relations of Perfect and Defect-Containing Waveguide Structures in Phononic Band Gap Materials
,”
Phys. Rev. B
,
68
(
2
), p.
024302
.10.1103/PhysRevB.68.024302
6.
Casadei
,
F.
,
Delpero
,
T.
,
Bergamini
,
A.
,
Ermanni
,
P.
, and
Ruzzene
,
M.
,
2012
, “
Piezoelectric Resonator Arrays for Tunable Acoustic Waveguides and Metamaterials
,”
J. Appl. Phys.
,
112
(
6
), p.
064902
.10.1063/1.4752468
7.
Faulkner
,
M.
, and
Hong
,
D.
,
1985
, “
Free Vibrations of a Mono-Coupled Periodic System
,”
J. Sound Vib.
,
99
(
1
), pp.
29
42
.10.1016/0022-460X(85)90443-2
8.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1995
,
Nonlinear Oscillations
,
Wiley
,
New York
.
9.
Nayfeh
,
A. H.
,
2008
,
Perturbation Methods
,
Wiley-VCH
, New York.
10.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
,
2008
,
Applied Nonlinear Dynamics
,
Wiley, New York
.
11.
Nayfeh
,
A. H.
, and
Pai
,
P. F.
,
2004
,
Linear and Nonlinear Structural Mechanics
,
Wiley
,
New York
.
12.
Morse
,
P. M.
, and
Ingard
,
K. U.
,
1987
,
Theoretical Acoustics
,
McGraw-Hill Book Co
, Princeton, NJ.
13.
Manktelow
,
K.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2013
, “
Comparison of Asymptotic and Transfer Matrix Approaches for Evaluating Intensity-Dependent Dispersion in Nonlinear Photonic and Phononic Crystals
,”
Wave Motion
,
50
(
3
), pp.
494
508
.10.1016/j.wavemoti.2012.12.009
14.
Manktelow
,
K.
,
Narisetti
,
R. K.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2012
, “
Finite-Element Based Perturbation Analysis of Wave Propagation in Nonlinear Periodic Structures
,”
Mech. Sys. Signal Proc.
,
39
(
1–2
), pp.
32
46
10.1016/j.ymssp.2012.04.015.
15.
Chakraborty
,
G.
, and
Mallik
,
A. K.
,
2001
, “
Dynamics of a Weakly Non-Linear Periodic Chain
,”
Int. J. Nonlin. Mech.
,
36
(
2
), pp.
375
389
.10.1016/S0020-7462(00)00024-X
16.
Hussein
,
M.
,
Leamy
,
M.
, and
Ruzzene
,
M.
,
2013
, “
Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress and Future Outlook
,”
ASME Appl. Mech. Rev.
(in press)10.1115/1.4026911.
17.
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
18.
Parmley
,
S.
,
Zobrist
,
T.
,
Clough
,
T.
,
Perez-Miller
,
A.
,
Makela
,
M.
, and
Yu
,
R.
,
1995
, “
Vibrational Properties of a Loaded String
,”
Am. J. Phys.
,
63
(
6
), p.
547
553
.10.1119/1.18089
19.
Parmley
,
S.
,
Zobrist
,
T.
,
Clough
,
T.
,
Perez-Miller
,
A.
,
Makela
,
M.
, and
Yu
,
R.
,
1995
, “
Phononic Band Structure in a Mass Chain
,”
Appl. Phys. Lett.
,
67
(
6
), pp.
777
779
.10.1063/1.115464
20.
Hopkins
, V
.
,
Krysac
,
L.
, and
Maynard
,
J.
,
1998
, “
Experimental Studies of Nonlinear Continuous Waves and Pulses in Disordered Media Showing Anderson Localization
,”
Phys. Rev. B
,
58
(
17
), p.
11377
.10.1103/PhysRevB.58.11377
21.
Daraio
,
C.
,
Nesterenko
, V
. F.
,
Herbold
,
E. B.
, and
Jin
,
S.
,
2005
, “
Strongly Nonlinear Waves in a Chain of Teflon Beads
,”
Phys. Rev. E
,
72
(
1
), p.
016603
.10.1103/PhysRevE.72.016603
22.
Cabaret
,
J.
,
Tournat
, V
.
, and
Béquin
,
P.
,
2012
, “
Amplitude Dependent Phononic Processes in a Diatomic Granular Chain in the Weakly Nonlinear Regime
,”
Phys. Rev. E
,
86
(
4
), p.
041305
.10.1103/PhysRevE.86.041305
23.
Yang
,
J.
,
Dunatunga
,
S.
, and
Daraio
,
C.
,
2012
, “
Amplitude-Dependent Attenuation of Compressive Waves in Curved Granular Crystals Constrained by Elastic Guides
,”
Acta Mech.
,
223
(
3
), pp.
1
14
10.1007/s00707-011-0568-x.
24.
Sugimoto
,
N.
,
1992
, “
Propagation of Nonlinear Acoustic Waves in a Tunnel With an Array of Helmholtz Resonators
,”
J. Fluid Mech.
,
244
, pp.
55
78
.10.1017/S0022112092002969
25.
Sugimoto
,
N.
, and
Horioka
,
T.
,
1995
, “
Dispersion Characteristics of Sound Waves in a Tunnel With an Array of Helmholtz Resonators
,”
J. Acoust. Soc. Am.
,
97
(
3
), pp.
1446
1459
.10.1121/1.412085
26.
Manktelow
,
K. L.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2011
, “
Multiple Scales Analysis of Wave-Wave Interactions in a Cubically Nonlinear Monoatomic Chain
,”
Nonlin. Dyn.
,
63
(
1–2
), pp.
193
203
.10.1007/s11071-010-9796-1
27.
Hladky-Hennion
,
A.-C.
, and
de Billy
,
M.
,
2007
, “
Experimental Validation of Band Gaps and Localization in a One-Dimensional Diatomic Phononic Crystal
,”
J. Acoust. Soc. Am.
,
122
(
5
), pp.
2594
2600
.10.1121/1.2779130
28.
Pierre
,
C.
,
1988
, “
Mode Localization and Eigenvalue Loci Veering Phenomena in Disordered Structures
,”
J. Sound Vib.
,
126
(
3
), pp.
485
502
.10.1016/0022-460X(88)90226-X
29.
Rushchitsky
,
J. J.
, and
Cattani
,
C.
,
2004
, “
Evolution Equations for Plane Cubically Nonlinear Elastic Waves
,”
Int. Appl. Mech.
,
40
(
1
), pp.
70
76
.10.1023/B:INAM.0000023812.41455.63
You do not currently have access to this content.