Abstract

Structures with geometric periodicity can present interesting dynamic properties like stop and pass frequency bands. In this case, the geometric periodicity has the effect of filtering the propagating waves in the structure, in a similar way to that of phononic crystals and metamaterials (non-homogeneous materials). Hence, by adopting such structures, we can design systems that present dynamic characteristics of interest, e.g., with minimum dynamic response in a given frequency range with large bandwidth. In the present work, we show that corrugated beams also present the dynamic properties of periodic structures due to their periodic geometry only (no need of changing mass or material properties along the beam). Two types of corrugated beams are studied analytically: beams with curved bumps of constant radii and beams with bumps composed of straight segments. The results show that, as we change the proportions of the bump, the natural frequencies change and tend to form large band gaps in the frequency spectrum of the beam. Such shifting of the natural frequencies is related to the coupling between longitudinal and transverse waves in the curved beam. The results also show that it is possible to predict the position and the limits of the first band gap (at least) as a function of the fundamental frequency of the straight beam (without bumps), irrespective of the total length of the corrugated beam.

References

1.
den Hartog
,
J. P.
,
1947
,
Mechanical Vibrations
,
McGraw-Hill Book Co.
,
New York
.
2.
Lu
,
Z.
,
Wang
,
Z.
,
Zhou
,
Y.
, and
Lu
,
X.
,
2018
, “
Nonlinear Dissipative Devices in Structural Vibration Control: A Review
,”
J. Sound. Vib.
,
423
, pp.
18
49
. 10.1016/j.jsv.2018.02.052
3.
Heckl
,
M. A.
,
1964
, “
Investigations on the Vibrations of Grillages and Other Simple Beam Structures
,”
J. Acoust. Soc. Am.
,
36
(
7
), pp.
1335
1343
. 10.1121/1.1919206
4.
Serrano
,
O.
,
Zaera
,
R.
, and
Fernandez-Saez
,
J.
,
2019
, “
On the Mechanism of Bandgap Formation in Beams With Periodic Arrangement of Beam-Like Resonators
,”
ASME J. Vib. Acoust.
,
141
(
6
), p.
064503
. 10.1115/1.4044863
5.
Chouvion
,
B.
,
Fox
,
C. H. J.
,
McWilliam
,
S.
, and
Popov
,
A. A.
,
2010
, “
In-Plane Free Vibration Analysis of Combined Ring-Beam Structural Systems by Wave Propagation
,”
J. Sound. Vib.
,
329
(
24
), pp.
5087
5104
. 10.1016/j.jsv.2010.05.023
6.
Xiao
,
Y.
,
Wen
,
J.
,
Wang
,
G.
, and
Wen
,
X.
,
2013
, “
Theoretical and Experimental Study of Locally Resonant and Bragg Band Gaps in Flexural Beams Carrying Periodic Arrays of Beam-Like Resonators
,”
ASME J. Vib. Acoust.
,
135
(
4
), p.
041006
. 10.1115/1.4024214
7.
Hajhosseini
,
M.
, and
Ebrahimi
,
S.
,
2019
, “
Analysis of Vibration Band Gaps in An Euler–Bernoulli Beam With Periodic Arrays of Meander-Shaped Beams
,”
J. Vib. Control
,
25
(
1
), pp.
41
51
. 10.1177/1077546318768995
8.
Junyi
,
L.
,
Ruffini
,
V.
, and
Balint
,
D.
,
2016
, “
Measuring the Band Structures of Periodic Beams Using the Wave Superposition Method
,”
J. Sound. Vib.
,
382
, pp.
158
178
. 10.1016/j.jsv.2016.07.005
9.
Timorian
,
S.
,
Petrone
,
G.
,
Rosa
,
S.
,
Franco
,
F.
,
Ouisse
,
M
, and
Bouhaddi
,
M.
,
2019
, “
Spectral Analysis and Structural Response of Periodic and Quasi-Periodic Beams
,”
J. Mech. Eng. Sci.
,
233
(
23–24
), pp.
7498
7512
. 10.1177/0954406219888948
10.
Syed
,
M.
, and
Bishay
,
P. L.
,
2019
, “
Analysis and Design of Periodic Beams for Vibration Attenuation
,”
J. Vib. Control
,
25
(
1
), pp.
228
239
. 10.1177/1077546318774436
11.
Prasad
,
R.
, and
Sarkar
,
A.
,
2019
, “
Broadband Vibration Isolation for Rods and Beams Using Periodic Structure Theory
,”
ASME J. Appl. Mech.
,
86
(
2
), p.
021004
. 10.1115/1.4042011
12.
Tang
,
L.
, and
Cheng
,
L.
,
2017
, “
Broadband Locally Resonant Band Gaps in Periodic Beam Structures With Embedded Acoustic Black Holes
,”
J. Appl. Phys.
,
121
(
19
), p.
194901
. 10.1063/1.4983459
13.
Pelat
,
A.
,
Gallot
,
T.
, and
Gautier
,
F.
,
2019
, “
On the Control of the First Bragg Band Gap in Periodic Continuously Corrugated Beam for Flexural Vibration
,”
J. Sound. Vib.
,
446
, pp.
249
262
. 10.1016/j.jsv.2019.01.029
14.
Alsaffar
,
Y.
,
Sassi
,
S.
, and
Baz
,
A.
,
2018
, “
Band Gap Characteristics of Periodic Gyroscopic Systems
,”
J. Sound. Vib.
,
435
, pp.
301
322
. 10.1016/j.jsv.2018.07.015
15.
Zak
,
A.
,
Krawczuk
,
M.
,
Redlarski
,
G.
,
Dolinski
,
L.
, and
Koziel
,
S.
,
2019
, “
A Three-Dimensional Periodic Beam for Vibroacoustic Isolation Purposes
,”
Mech. Syst. Signal Proc.
,
130
, pp.
524
544
. 10.1016/j.ymssp.2019.05.033
16.
Matlack
,
K. H.
,
Bauhofer
,
A.
,
Krödel
,
S.
,
Palermo
,
A.
, and
Daraio
,
C.
,
2016
, “
Composite 3D-Printed Metastructures for Low-Frequency and Broadband Vibration Absorption
,”
Proc. Natl. Acad. Sci. U. S. A.
,
113
(
30
), pp.
8386-8390
. 10.1073/pnas.1600171113
17.
Beli
,
D.
,
Fabro
,
A. T.
,
Ruzzene
,
M.
, and
Arruda
,
J. R. F.
,
2019
, “
Wave Attenuation and Trapping in 3D Printed Cantilever-in-Mass Metamaterials With Spatially Correlated Variability
,”
Sci. Rep.
,
9
, p.
5617
. 10.1038/s41598-019-41999-0
18.
Elmadih
,
W.
,
Chronopoulos
,
D.
,
Syam
,
W. P.
,
Maskery
,
I.
,
Meng
,
H.
, and
Leach
,
R. K.
,
2019
, “
Three-dimensional Resonating Metamaterials for Low-Frequency Vibration Attenuation
,”
Sci. Rep.
,
9
, p.
11503
. 10.1038/s41598-019-47644-0
19.
Loyau
,
T.
,
Weinachter
,
P.
,
Rebillard
,
E.
, and
Guyader
,
J. L.
,
1997
, “
Experimental Study of Vibration Response Dispersion Between Structures
,”
J. Sound. Vib.
,
203
(
5
), pp.
894
898
. 10.1006/jsvi.1996.0880
20.
Robin
,
G.
,
Jrad
,
M.
,
Mathieu
,
N.
,
Daouadji
,
A.
, and
Dayaa
,
E. M.
,
2016
, “
Vibration Analysis of Corrugated Beams: the Effects of Temperature and Corrugation Shape
,”
Mech. Res. Commun.
,
71
, pp.
1
6
. 10.1016/j.mechrescom.2015.11.002
21.
Wang
,
N.
,
Zhang
,
Z.
, and
Zhang
,
X.
,
2019
, “
Stiffness Analysis of Corrugated Flexure Beam Using Stiffness Matrix Method
,”
J. Mech. Eng. Sci.
,
233
(
5
), pp.
1818
1827
. 10.1177/0954406218772002
22.
Kim
,
I-H.
,
Jin
,
S-S.
,
Jang
,
S-J.
, and
Jung
,
H-J.
,
2014
, “
A Performance-Enhanced Energy Harvester for Low Frequency Vibration Utilizing a Corrugated Cantilevered Beam
,”
Smart Mater. Struct.
,
23
(
3
), p.
037002
. 10.1088/0964-1726/23/3/037002
23.
Doyle
,
J. F.
,
1997
,
Wave Propagation in Structures
,
Springer
,
New York
.
24.
Nicoletti
,
R.
,
2020
, “
On the Natural Frequencies of Simply Supported Beams Curved in Mode Shapes
,”
J. Sound. Vib.
,
485
, p.
115597
. 10.1016/j.jsv.2020.115597
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