The paper investigates wave dispersion in two-dimensional, weakly nonlinear periodic lattices. A perturbation approach, originally developed for one-dimensional systems and extended herein, allows for closed-form determination of the effects nonlinearities have on dispersion and group velocity. These expressions are used to identify amplitude-dependent bandgaps, and wave directivity in the anisotropic setting. The predictions from the perturbation technique are verified by numerically integrating the lattice equations of motion. For small amplitude waves, excellent agreement is documented for dispersion relationships and directivity patterns. Further, numerical simulations demonstrate that the response in anisotropic nonlinear lattices is characterized by amplitude-dependent “dead zones.”
Issue Section:
Research Papers
References
1.
Brillouin
, L.
, 1946, Wave Propagation in Periodic Structures
, Dover
, New York
.2.
Mead
, D. J.
, and Parthan
, S.
, 1979, “Free Wave Propagation in Two-Dimensional Periodic Plates
,” J. Sound Vib.
, 64
(3
), pp. 325
–348
. 3.
Mead
, D. J.
, Zhu
, D. C.
, and Bardell
, N. S.
, 1988, “Free Vibration of an Orthogonally Stiffened Flat Plate
,” J. Sound Vib.
, 127
(1
), pp. 19
–48
. 4.
Mace
, B.
, and Manconi
, E.
, 2008, “Modelling Wave Propagation in Two-Dimensional Structures Using Finite Element Analysis
,” J. Sound Vib.
, 318
(4-5
), pp. 884
–902
. 5.
Mead
, D. J.
, 1996, “Wave Propagation in Continous Periodic Structures: Research Contributions from Southampton 1964–1995
,” J. Sound Vib.
, 190
(3
), pp. 495
–524
. 6.
Ruzzene
, M.
, Scarpa
, F.
, and Soranna
, F.
, 2003, “Wave Beaming Effects in Two-Dimensional Cellular Structures
,” Smart Mater. Struct.
, 12
(3
), pp. 363
–372
. 7.
Ruzzene
, M.
, and Tsopelas
, P.
, 2003, “Control of Wave Propagation in Sandwich Plate Rows with Perioidc Honeycomb Core
,” J. Eng. Mech.
, 129
(9
), pp. 975
–986
. 8.
Phani
, A. S.
, Woodhouse
, J.
, and Fleck
, N. A.
, 2006, “Wave Propagation in Two-Dimensional Periodic Lattices
,” J. Acoust. Soc. Am.
, 119
(4
), pp. (1995
). 9.
Kohrs
, T.
, and Petersson
, B. a. T.
, 2009, “Wave Beaming and Wave Propagation in Light Weight Plates with Truss-Like Cores
,” J. Sound Vib.
, 321
(1-2
), pp. 137
–165
. 10.
Langley
, R. S.
, 1996, “The Response of Two-Dimensional Periodic Structures to Point Harmonic Forcing
,” J. Sound Vib.
, 197
(4
), pp. 447
–469
. 11.
Langley
, R. S.
, Bardell
, N. S.
, and Ruivo
, H. M.
, 1997, “The Response of Two-Dimensional Periodic Structures to Harmonic Point Loading: A Theoretical and Experimental Study of a Beam Grillage
,” J. Sound Vib.
, 207
(4
), pp. 521
–535
. 12.
Bullough
, R. K.
, and Caudrey
, P. J.
, 1980, Solitons
, Springer
, Berlin
.13.
Aceves
, A.
, 2007, “Localization and Trapping of Light in One- and Two-Dimensional Nonlinear Periodic Structures
,” Wave Motion
, 45
(1-2
), pp. 48
–58
. 14.
Babaoglu
, C.
, and Erbay
, S.
, 2001, “Two-Dimensional Wave Propagation in a Generalized Elastic Solid
,” Chaos, Solitons Fractals
, 12
(2
), pp. 381
–389
. 15.
Sreelatha
, K. S.
, and Babu Joseph
, K.
, 2000, “Wave Propagation through a 2D Lattice
,” Chaos Solitons Fractals
, 11
(5
), pp. 711
–719
. 16.
Duan
, W. S.
, Shi
, Y.
, Zhang
, L.
, Lin
, M.-M.
, and Lv
, K.
, 2005, “Coupled Nonlinear Waves in Two-Dimensional Lattice
,” Chaos Solitons Fractals
, 23
(3
), pp. 957
–962
.17.
Lü
, B.-B.
, and Tian
, Q.
, 2010, “Different Kinds of Discrete Breathers in Three Types of One-Dimensional Models
,” Commun. Theor. Phys.
, 54
(4
), pp. 728
. 18.
Feng
, B.
, and Kawahara
, T.
, 2007, “Discrete Breathers in Two-Dimensional Nonlinear Lattices
,” Wave Motion
, 45
(1-2
), pp. 68
–82
. 19.
Franchini
, A.
, and et al., 2002, “Theory of Intrinsic Localized Modes in Diatomic Chains: Beyond the Rotating Wave Approximation
,” J. Phys.: Condens. Matter
, 14
(2
), pp. 145
. 20.
He
, and J.
, 2004, “The Homotopy Perturbation Method for Nonlinear Oscillators with Discontinuities
,” Appl. Math. Comput.
, 151
(1
), pp. 287
–292
. 21.
He
, J.-H.
, 2002, “Modified Lindstedt–Poincare Methods for Some Strongly Non-Linear Oscillations: Part II: A New Transformation
,” Int. J. Non-Linear Mech.
, 37
(2
), pp. 315
–320
. 22.
Narisetti
, R. K.
, Leamy
, M. J.
, and Ruzzene
, M.
, 2010, “A Perturbation Approach for Predicting Wave Propagation in One Dimensional Nonlinear Periodic Structures
,” J. Vib. Acoust.
, 132
(3
), pp. 1001
–1012
. 23.
Nayfeh
, A.
, and Mook
, D.
, 1996, Nonlinear Oscillations
, Wiley
, New York
.24.
Spadoni
, A.
Ruzzene
, M.
Gonella
, S.
, and Scarpa
, F.
, 2009, “Phononic Properties of Hexagonal Chiral Lattices
,” Wave Motion
, 46
(7
), pp. 435
–450
. 25.
Wolfe
, J. P.
, 1998, Imaging Phonons: Acoustic Wave Propagation in Solids
, Cambridge University Press
, Cambridge
.Copyright © 2011
by American Society of Mechanical Engineers
You do not currently have access to this content.
