Abstract
In this paper, flexural wave propagation, attenuation, and reflection through finite number of rigid elastic combined metabeam (RECM) elements sandwiched between two Euler Bernoulli beams has been studied, implementing the spectral element, inverse Fourier transform, and transfer matrix method. Spectral element has been formulated for the unit representative cell of RECM employing the rigid body dynamics. Governing dimensionless parameters are identified. Furthermore, the sensitivity analysis has been carried out to comprehend the influence of non-dimensional parameters, such as mass ratio, length ratio, and rotary inertia ratio on the attenuation profile. Rotary inertia of rigid body produces local resonance (LR) band, which may abridge the gap between the two Bragg scattering (BS) bands and results in an ultra-wide stop band for the specific combination of governing non-dimensional parameters. A total of 164% normalized attenuation band is possible to obtain in RECM. Natural frequencies for the finite RECM have also been evaluated from the global spectral element matrix and observed that some natural frequencies lie in the attenuation band. Therefore, the level of attenuation near that natural frequencies is significantly less and cannot be identified from the dispersion diagram of the infinite RECM.
Issue Section:
Research Papers
References
1.
Mead
, D.
, 1973
, “A General Theory of Harmonic Wave Propagation in Linear Periodic Systems With Multiple Coupling
,” J. Sound Vib.
, 27
(2
), pp. 235
–260
. 2.
Mead
, D.
, 1996
, “Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton, 1964–1995
,” J. Sound Vib.
, 190
(3
), pp. 495
–524
. 3.
SenGupta
, G.
, 1980
, “Vibration of Periodic Structures
,” Shock Vib. Dig.
, 12
(3
), pp. 17
–31
. 4.
Roy
, A.
, and Plunkett
, R.
, 1986
, “Wave Attenuation in Periodic Structures
,” J. Sound Vib.
, 104
(3
), pp. 395
–410
. 5.
Orris
, R. M.
, and Petyt
, M.
, 1974
, “A Finite Element Study of Harmonic Wave Propagation in Periodic Structures
,” J. Sound Vib.
, 33
(2
), pp. 223
–236
. 6.
Mace
, B.
, 1984
, “Wave Reflection and Transmission in Beams
,” J. Sound Vib.
, 97
(2
), pp. 237
–246
. 7.
Brillouin
, L.
, 1953
, Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices
, Dover
, New York
.8.
Mead
, D. J.
, 1970
, “Free Wave Propagation in Periodically Supported, Infinite Beams
,” J. Sound Vib.
, 11
(2
), pp. 181
–197
. 9.
Banerjee
, A.
, Das
, R.
, and Calius
, E. P.
, 2019
, “Waves in Structured Mediums or Metamaterials: A Review
,” Arch. Comput. Methods Eng.
, 26
(4
), pp. 1029
–1058
. 10.
Prasad
, R.
, and Banerjee
, A.
, 2021
, “Influence of Conicity on the Free Wave Propagation in Symmetric Tapered Periodic Beam
,” Mech. Res. Commun.
, 111
, p. 103655
. 11.
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
. 12.
Yuksel
, O.
, and Yilmaz
, C.
, 2020
, “Realization of an Ultrawide Stop Band in a 2-d Elastic Metamaterial With Topologically Optimized Inertial Amplification Mechanisms
,” Int. J. Solids Struct.
, 203
, pp. 138
–150
. 13.
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
. 14.
Ozmutlu
, A.
, Ebrahimian
, M.
, and Todorovska
, M. I.
, 2018
, “Wave Propagation in Buildings as Periodic Structures: Timoshenko Beam With Rigid Floor Slabs Model
,” J. Eng. Mech.
, 144
(4
), p. 04018010
. 15.
Frandsen
, N. M.
, Bilal
, O. R.
, Jensen
, J. S.
, and Hussein
, M. I.
, 2016
, “Inertial Amplification of Continuous Structures: Large Band Gaps From Small Masses
,” J. Appl. Phys.
, 119
(12
), p. 124902
. 16.
Yu
, D.
, Wen
, J.
, Shen
, H.
, Xiao
, Y.
, and Wen
, X.
, 2012
, “Propagation of Flexural Wave in Periodic Beam on Elastic Foundations
,” Phys. Lett. A
, 376
(4
), pp. 626
–630
. 17.
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
. 18.
Banerjee
, A.
, 2020
, “Non-Dimensional Analysis of the Elastic Beam Having Periodic Linear Spring Mass Resonators
,” Meccanica
, 55
(5
), pp. 1181
–1191
. 19.
Banerjee
, A.
, 2020
, “Influence of the Torsional Vibration of the Periodically Attached Perpendicular Beam Resonator on the Flexural Band of a Euler–Bernoulli Beam
,” Phys. Lett. A
, 384
(29
), p. 126757
. 20.
Wang
, T.
, Sheng
, M.-P.
, and Qin
, Q.-H.
, 2016
, “Multi-Flexural Band Gaps in an Euler–Bernoulli Beam With Lateral Local Resonators
,” Phys. Lett. A
, 380
(4
), pp. 525
–529
. 21.
Sugino
, C.
, and Erturk
, A.
, 2018
, “Analysis of Multifunctional Piezoelectric Metastructures for Low-Frequency Bandgap Formation and Energy Harvesting
,” J. Phys. D: Appl. Phys.
, 51
(21
), p. 215103
. 22.
Banerjee
, A.
, 2021
, “Flexural Waves in Graded Metabeam Lattice
,” Phys. Lett. A
, 388
, p. 127057
. 23.
Hu
, G.
, Austin
, A. C.
, Sorokin
, V.
, and Tang
, L.
, 2021
, “Metamaterial Beam With Graded Local Resonators for Broadband Vibration Suppression
,” Mech. Syst. Signal Process.
, 146
, p. 106982
. 24.
Gopalakrishnan
, S.
, and Doyle
, J.
, 1994
, “Wave Propagation in Connected Waveguides of Varying Cross-Section
,” J. Sound Vib.
, 175
(3
), pp. 347
–363
. 25.
Lee
, S.-K.
, Mace
, B.
, and Brennan
, M.
, 2007
, “Wave Propagation, Reflection and Transmission in Non-uniform One-Dimensional Waveguides
,” J. Sound Vib.
, 304
(1–2
), pp. 31
–49
. 26.
Manohar
, C.
, and Adhikari
, S.
, 1998
, “Dynamic Stiffness of Randomly Parametered Beams
,” Probab. Eng. Mech.
, 13
(1
), pp. 39
–51
. 27.
Lee
, U.
, 2009
, Spectral Element Method in Structural Dynamics
, John Wiley & Sons
, Singapore
.28.
Jang
, I.
, and Lee
, U.
, 2012
, “Spectral Element Analysis of the Axial-Bending-Shear Coupled Vibrations of Composite Timoshenko Beams
,” J. Compos. Mater.
, 46
(22
), pp. 2811
–2828
. 29.
Low
, K.
, 1998
, “On the Eigenfrequencies for Mass Loaded Beams Under Classical Boundary Conditions
,” J. Sound Vib.
, 215
(2
), pp. 381
–389
. 30.
Banerjee
, J.
, and Sobey
, A.
, 2003
, “Further Investigation Into Eigenfrequencies of a Two-Part Beam–Mass System
,” J. Sound Vib.
, 265
(4
), pp. 899
–908
. 31.
Obradović
, A.
, Šalinić
, S.
, Trifković
, D. R.
, Zorić
, N.
, and Stokić
, Z.
, 2015
, “Free Vibration of Structures Composed of Rigid Bodies and Elastic Beam Segments
,” J. Sound Vib.
, 347
, pp. 126
–138
.32.
Tomović
, A.
, Šalinić
, S.
, Obradović
, A.
, Grbović
, A.
, and Milovančević
, M.
, 2020
, “Closed-Form Solution for the Free Axial-Bending Vibration Problem of Structures Composed of Rigid Bodies and Elastic Beam Segments
,” Appl. Math. Model.
, 77
(part 2
), pp. 1148
–1167
.33.
Liu
, X.
, Sun
, C.
, Ranjan Banerjee
, J.
, Dan
, H.-C.
, and Chang
, L.
, 2021
, “An Exact Dynamic Stiffness Method for Multibody Systems Consisting of Beams and Rigid-Bodies
,” Mech. Syst. Signal Process.
, 150
, p. 107264
. 34.
Wu
, J.-J.
, 2011
, “Use of the Elastic-and-Rigid-Combined Beam Element for Dynamic Analysis of a Two-Dimensional Frame With Arbitrarily Distributed Rigid Beam Segments
,” Appl. Math. Model.
, 35
(3
), pp. 1240
–1251
. 35.
Mead
, D.
, 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
. 36.
Fu
, Z.-F.
, and He
, J.
, 2001
, Modal Analysis
, Elsevier
, Oxford, UK
.37.
Banerjee
, A.
, Das
, R.
, and Calius
, E. P.
, 2016
, “A New Approach for Determinationof the Attenuation Bandwidth of a Resonating Metamaterial,” Applied Mechanics and Materials
, Y.
Gu
, H.
Guan
, E.
Sauret
, S.
Saha
, H.
Zhan
, and R.
Persky
, eds., Vol. 846
, Trans Tech Publications
, Switzerland
, pp. 264
–269
.38.
Davis
, B. L.
, Tomchek
, A. S.
, Flores
, E. A.
, Liu
, L.
, and Hussein
, M. I.
, 2011
, “Analysis of Periodicity Termination in Phononic Crystals
,” ASME International Mechanical Engineering Congress and Exposition
, Denver, CO
, Nov. 11–17
, Vol. 54945, pp. 973
–977
.39.
Doyle
, J. F.
, 2012
, Wave Propagation in Structures: An FFT-Based Spectral Analysis Methodology
, Springer Science & Business Media
, New York
.40.
Huang
, H.
, and Sun
, C.
, 2011
, “Theoretical Investigation of the Behavior of an Acoustic Metamaterial With Extreme Young’s Modulus
,” J. Mech. Phys. Solids
, 59
(10
), pp. 2070
–2081
. Copyright © 2022 by ASME
You do not currently have access to this content.
