In this paper, we present a design of locally resonant (LR) beams using periodic arrays of beam-like resonators (or beam-like vibration absorbers) attached to a thin homogeneous beam. The main purpose of this work is twofold: (i) providing a theoretical characterization of the proposed LR beams, including the band gap behavior of infinite systems and the vibration transmittance of finite structures, and (ii) providing experimental evidence of the associated band gap properties, especially the coexistence of LR and Bragg band gaps, and their evolution with tuned local resonance. For the first purpose, an analytical method based on the spectral element formulations is presented, and then an in-depth numerical study is performed to examine the band gap effects. In particular, explicit formulas are provided to enable an exact calculation of band gaps and an approximate prediction of band gap edges. For the second purpose, we fabricate several LR beam specimens by mounting 16 equally spaced resonators onto a free-free host beam. These specimens use the same host beam, but the resonance frequencies of the resonators on each beam are different. We further measure the vibration transmittances of these specimens, which give evidence of three interesting band gap phenomena: (i) transition between LR and Bragg band gaps; (ii) near-coupling effect of the local resonance and Bragg scattering; and (iii) resonance frequency of local resonators outside of the LR band gap.

References

References
1.
Mead
,
D. J.
,
1970
, “
Free Wave Propagation in Periodically Supported, Infinite Beams
,”
J. Sound Vib.
,
11
(
2
), pp.
181
197
.10.1016/S0022-460X(70)80062-1
2.
Mead
,
D. J.
,
1975
, “
Wave Propagation and Natural Modes in Periodic Systems: I. Mono-Coupled Systems
,”
J. Sound Vib.
,
40
(
1
), pp.
1
18
.10.1016/S0022-460X(75)80227-6
3.
Mead
,
D. J.
,
1975
, “
Wave Propagation and Natural Modes in Periodic Systems: II. Multi-Coupled Systems, With and Without Damping
,”
J. Sound Vib.
,
40
(
1
), pp.
19
39
.10.1016/S0022-460X(75)80228-8
4.
Mead
,
D. J.
,
1996
, “
Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton, 1964-1995
,”
J. Sound Vib.
,
190
(
3
), pp.
495
524
.10.1006/jsvi.1996.0076
5.
Baz
,
A.
,
2001
, “
Active Control of Periodic Structures
,”
ASME J. Vib. Acoust.
,
123
(
4
), pp.
472
479
.10.1115/1.1399052
6.
Jensen
,
J. S.
,
2003
, “
Phononic Band Gaps and Vibrations in One- and Two-Dimensional Mass-Spring Structures
,”
J. Sound Vib.
,
266
(
5
), pp.
1053
1078
.10.1016/S0022-460X(02)01629-2
7.
Lazarov
,
B. S.
, and
Jensen
,
J. S.
,
2007
, “
Low-Frequency Band Gaps in Chains With Attached Non-Linear Oscillators
,”
Int. J. Nonlinear Mech.
,
42
(
10
), pp.
1186
1193
.10.1016/j.ijnonlinmec.2007.09.007
8.
Ruzzene
,
M.
, and
Baz
,
A.
,
2000
, “
Control of Wave Propagation in Periodic Composite Rods Using Shape Memory Inserts
,”
ASME J. Vibr. Acoust.
,
122
(
2
), pp.
151
159
.10.1115/1.568452
9.
Thorp
,
O.
,
Ruzzene
,
M.
, and
Baz
,
A.
,
2001
, “
Attenuation and Localization of Wave Propagation in Rods With Periodic Shunted Piezoelectric Patches
,”
Smart Mater. Struct.
,
10
(
5
), pp.
979
989
.10.1088/0964-1726/10/5/314
10.
Hussein
,
M. I.
,
Hulbert
,
G. M.
, and
Scott
,
R. A.
,
2006
, “
Dispersive Elastodynamics of 1D Banded Materials and Structures: Analysis
,”
J. Sound Vib.
,
289
(
4-5
), pp.
779
806
.10.1016/j.jsv.2005.02.030
11.
Hussein
,
M. I.
,
Hulbert
,
G. M.
, and
Scott
,
R. A.
,
2007
, “
Dispersive Elastodynamics of 1D Banded Materials and Structures: Design
,”
J. Sound Vib.
,
307
(
3–5
), pp.
865
893
.10.1016/j.jsv.2007.07.021
12.
Romeo
,
F.
, and
Luongo
,
A.
,
2003
, “
Vibration Reduction in Piecewise Bi-coupled Periodic Structures
,”
J. Sound Vib.
,
268
, pp.
601
615
.10.1016/S0022-460X(03)00375-4
13.
Langley
,
R. S.
,
1996
, “
The Response of Two-Dimensional Periodic Structures to Point Harmonic Forcing
,”
J. Sound Vib.
,
197
(
4
), pp.
447
469
.10.1006/jsvi.1996.0542
14.
Ruzzene
,
M.
,
Scarpa
,
F.
, and
Soranna
,
F.
,
2003
, “
Wave Beaming Effects in Two-Dimensional Cellular Structures
,”
Smart Mater. Struct.
,
12
(
3
), pp.
363
372
.10.1088/0964-1726/12/3/307
15.
Phani
,
A. S.
,
Woodhouse
,
J.
, and
Fleck
,
N. A.
,
2006
, “
Wave Propagation in Two-Dimensional Periodic Lattices
,”
J. Acoust. Soc. Am.
,
119
, pp.
1995
2005
.10.1121/1.2179748
16.
Leamy
,
M. J.
,
2012
, “
Exact Wave-Based Bloch Analysis Procedure for Investigating Wave Propagation in Two-Dimensional Periodic Lattices
,”
J. Sound Vib.
,
331
(
7
), pp.
1580
1596
.10.1016/j.jsv.2011.11.023
17.
Kushwaha
,
M. S.
,
Halevi
,
P.
,
Dobrzynski
,
L.
, and
Djafari-Rouhani
,
B.
,
1993
, “
Acoustic Band Structure of Periodic Elastic Composites
,”
Phys. Rev. Lett.
,
71
(
13
), pp.
2022
2025
.10.1103/PhysRevLett.71.2022
18.
Liu
,
Z.
,
Zhang
,
X.
,
Mao
,
Y.
,
Zhu
,
Y. 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
19.
Sigmund
,
O.
, and
Jensen
,
J. S.
,
2003
, “
Systematic Design of Phononic Band-Gap Materials and Structures by Topology Optimization
,”
Philos. Trans. R. Soc. Lond. A
,
361
(
1806
), pp.
1001
1019
.10.1098/rsta.2003.1177
20.
Halkjaer
,
S.
,
Sigmund
,
O.
, and
Jensen
,
J. S.
,
2006
, “
Maximizing Band Gaps in Plate Structures
,”
Struct. Multidiscip. Optim.
,
32
(
4
), pp.
263
275
.10.1007/s00158-006-0037-7
21.
Diaz
,
A. R.
,
Haddow
,
A. G.
, and
Ma
,
L.
,
2005
, “
Design of Band-Gap Grid Structures
,”
Struct. Multidiscip. Optim.
,
29
(
6
), pp.
418
431
.10.1007/s00158-004-0497-6
22.
Policarpo
,
H.
,
Neves
,
M. M.
, and
Ribeiro
,
A. M.
,
2010
, “
Dynamical Response of a Multi-Laminated Periodic Bar: Analytical, Numerical and Experimental Study
,”
Shock Vib.
,
17
(
4
), pp.
521
535
.10.3233/SAV-2010-0545
23.
Hussein
,
M. I.
,
Hamza
,
K.
,
Hulbert
,
G.
,
Scott
,
R.
, and
Saitou
,
K.
,
2006
, “
Multiobjective Evolutionary Optimization of Periodic Layered Materials for Desired Wave Dispersion Characteristics
,”
Struct. Multidiscip. Optim.
,
31
(
1
), pp.
60
75
.10.1007/s00158-005-0555-8
24.
Bilal
,
O. R.
, and
Hussein
,
M. I.
,
2011
, “
Ultrawide Phononic Band Gap for Combined In-Plane and Out-of-Plane Waves
,”
Phys. Rev. E
,
84
(
6
), p.
065701
.10.1103/PhysRevE.84.065701
25.
Goffaux
,
C.
,
Sanchez-Dehesa
,
J.
,
Yeyati
,
A. L.
,
Lambin
,
P.
,
Khelif
,
A.
,
Vasseur
,
J. O.
, and
Djafari-Rouhani
,
B.
,
2002
, “
Evidence of Fano-Like Interference Phenomena in Locally Resonant Materials
,”
Phys. Rev. Lett.
,
88
(
22
), p.
225502
.10.1103/PhysRevLett.88.225502
26.
Wang
,
G.
,
Wen
,
X.
,
Wen
,
J.
,
Shao
,
L.
, and
Liu
,
Y.
,
2004
, “
Two-Dimensional Locally Resonant Phononic Crystals With Binary Structures
,”
Phys. Rev. Lett.
,
93
(
15
), p.
154302
.10.1103/PhysRevLett.93.154302
27.
Wang
,
G.
,
Wen
,
X.
,
Wen
,
J.
, and
Liu
,
Y.
,
2006
, “
Quasi-One-Dimensional Periodic Structure With Locally Resonant Band Gap
,”
ASME J. Appl. Mech.
,
73
(
1
), pp.
167
170
.10.1115/1.2061947
28.
Xiao
,
Y.
,
Wen
,
J.
, and
Wen
,
X.
,
2012
, “
Longitudinal Wave Band Gaps in Metamaterial-Based Elastic Rods Containing Multi-Degree-of-Freedom Resonators
,”
New J. Phys.
,
14
(
3
), p.
033042
.10.1088/1367-2630/14/3/033042
29.
Yu
,
D.
,
Liu
,
Y.
,
Wang
,
G.
,
Zhao
,
H.
, and
Qiu
,
J.
,
2006
, “
Flexural Vibration Band Gaps in Timoshenko Beams With Locally Resonant Structures
,”
J. Appl. Phys.
,
100
(
12
), p.
124901
.10.1063/1.2400803
30.
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
31.
Xiao
,
Y.
,
Wen
,
J.
, and
Wen
,
X.
,
2012
, “
Broadband Locally Resonant Beams Containing Multiple Periodic Arrays of Attached Resonators
,”
Phys. Lett. A
,
376
(
16
), pp.
1384
1390
.10.1016/j.physleta.2012.02.059
32.
Xiao
,
Y.
,
Wen
,
J.
,
Yu
,
D.
, and
Wen
,
X.
,
2013
, “
Flexural Wave Propagation in Beams With Periodically Attached Vibration Absorbers: Band-Gap Behavior and Band Formation Mechanisms
,”
J. Sound Vib.
,
332
(
4
), pp.
867
893
.10.1016/j.jsv.2012.09.035
33.
Oudich
,
M.
,
Senesi
,
M.
,
Assouar
,
M. B.
,
Ruzenne
,
M.
,
Sun
,
J. H.
,
Vincent
,
B.
,
Hou
,
Z.
, and
Wu
,
T. T.
,
2011
, “
Experimental Evidence of Locally Resonant Sonic Band Gap in Two-Dimensional Phononic Stubbed Plates
,”
Phys. Rev. B
,
84
(
16
), p.
165136
.10.1103/PhysRevB.84.165136
34.
Xiao
,
Y.
,
Wen
,
J.
, and
Wen
,
X.
,
2012
, “
Flexural Wave Band Gaps in Locally Resonant Thin Plates With Periodically Attached Spring-Mass Resonators
,”
J. Phys. D: Appl. Phys.
,
45
(
19
), p.
195401
.10.1088/0022-3727/45/19/195401
35.
Xiao
,
Y.
,
Wen
,
J.
, and
Wen
,
X.
,
2012
, “
Sound Transmission Loss of Metamaterial-Based Thin Plates With Multiple Subwavelength Arrays of Attached Resonators
,”
J. Sound Vib.
,
331
(
25
), pp.
5408
5423
.10.1016/j.jsv.2012.07.016
36.
Xiao
,
Y.
,
Mace
,
B. R.
,
Wen
,
J.
, and
Wen
,
X.
,
2011
, “
Formation and Coupling of Band Gaps in a Locally Resonant Elastic System Comprising a String With Attached Resonators
,”
Phys. Lett. A
,
375
(
12
), pp.
1485
1491
.10.1016/j.physleta.2011.02.044
37.
Jacquot
,
R. G.
, and
Foster
,
J. E.
,
1977
, “
Optimal Cantilever Dynamic Vibration Absorbers
,”
ASME J. Eng. Ind.
,
99
(
1
), pp.
138
141
.10.1115/1.3439127
38.
Strasberg
,
M.
, and
Feit
,
D.
,
1996
, “
Vibration Damping of Large Structures Induced by Attached Small Resonant Structures
,”
J. Acoust. Soc. Am.
,
99
(
1
), pp.
335
344
.10.1121/1.414545
39.
Brennan
,
M. J.
,
1997
, “
Characteristics of a Wideband Vibration Neutralizer
,”
Noise Control Eng. J.
,
45
(
5
), pp.
201
207
.10.3397/1.2828441
40.
Sun
,
J. Q.
,
Jolly
,
M. R.
, and
Norris
,
M. A.
,
1995
, “
Passive, Adaptive and Active Tuned Vibration Absorbers—A Survey
,”
ASME J. Mech. Des.
,
117
, pp.
234
242
.10.1115/1.2836462
41.
Thompson
,
D. J.
,
2008
, “
A Continuous Damped Vibration Absorber to Reduce Broad-Band Wave Propagation in Beams
,”
J. Sound Vib.
,
311
(
3-5
), pp.
824
842
.10.1016/j.jsv.2007.09.038
42.
El-Khatib
,
H. M.
,
Mace
,
B. R.
, and
Brennan
,
M. J.
,
2005
, “
Suppression of Bending Waves in a Beam Using a Tuned Vibration Absorber
,”
J. Sound Vib.
,
288
(
4-5
), pp.
1157
1175
.10.1016/j.jsv.2005.01.024
43.
Graff
,
K. F.
,
1975
,
Wave Motion in Elastic Solids
,
Oxford University
,
London
.
44.
Mead
,
D. J.
,
1986
, “
A New Method of Analyzing Wave Propagation in Periodic Structures: Applications to Periodic Timoshenko Beams and Stiffened Plates
,”
J. Sound Vib.
,
104
(
1
), pp.
9
27
.10.1016/S0022-460X(86)80128-6
45.
Lee
,
U.
,
2009
,
Spectral Element Method in Structural Dynamics
,
Wiley
,
Singapore
.
46.
Brillouin
,
L.
,
1946
,
Wave Propagation in Periodic Structures
,
Dover
,
New York
.
47.
Ruzzene
,
M.
, and
Scarpa
,
F.
,
2003
, “
Control of Wave Propagation in Sandwich Beams With Auxetic Core
,”
J. Intell. Mater. Syst. Struct.
,
14
(
7
), pp.
443
453
.10.1177/1045389X03035515
48.
Davis
,
B. L.
,
Tomchek
,
A. S.
,
Flores
,
E. A.
,
Liu
,
L.
, and
Hussein
,
M. I.
,
2011
, “
Analysis of Periodicity Termination in Phononic Crystals
,”
Proceedings of the ASME 2011 International Mechanical Engineering Congress & Exposition
,
Denver
, CO, November 11–17,
ASME
Paper No. IMECE2011-65666.10.1115/IMECE2011-65666
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