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Research Papers

Theoretical and Experimental Study of Locally Resonant and Bragg Band Gaps in Flexural Beams Carrying Periodic Arrays of Beam-Like Resonators

[+] Author and Article Information
Yong Xiao

e-mail: xiaoy@vip.sina.com

Jihong Wen

e-mail: wenjihong@vip.sina.com

Gang Wang

e-mail: wangg@nudt.edu.cn

Xisen Wen

e-mail: wenxs@vip.sina.com
Vibration and Acoustics Research Group,
Laboratory of Science and Technology on Integrated Logistics Support,
College of Mechatronics and Automation,
National University of Defense Technology,
Changsha 410073, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 6, 2012; final manuscript received April 8, 2013; published online June 6, 2013. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 135(4), 041006 (Jun 06, 2013) (17 pages) Paper No: VIB-12-1140; doi: 10.1115/1.4024214 History: Received May 06, 2012; Revised April 08, 2013

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.

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References

Mead, D. J., 1970, “Free Wave Propagation in Periodically Supported, Infinite Beams,” J. Sound Vib., 11(2), pp. 181–197. [CrossRef]
Mead, D. J., 1975, “Wave Propagation and Natural Modes in Periodic Systems: I. Mono-Coupled Systems,” J. Sound Vib., 40(1), pp. 1–18. [CrossRef]
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. [CrossRef]
Mead, D. J., 1996, “Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton, 1964-1995,” J. Sound Vib., 190(3), pp. 495–524. [CrossRef]
Baz, A., 2001, “Active Control of Periodic Structures,” ASME J. Vib. Acoust., 123(4), pp. 472–479. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Romeo, F., and Luongo, A., 2003, “Vibration Reduction in Piecewise Bi-coupled Periodic Structures,” J. Sound Vib., 268, pp. 601–615. [CrossRef]
Langley, R. S., 1996, “The Response of Two-Dimensional Periodic Structures to Point Harmonic Forcing,” J. Sound Vib., 197(4), pp. 447–469. [CrossRef]
Ruzzene, M., Scarpa, F., and Soranna, F., 2003, “Wave Beaming Effects in Two-Dimensional Cellular Structures,” Smart Mater. Struct., 12(3), pp. 363–372. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
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. [CrossRef]
Halkjaer, S., Sigmund, O., and Jensen, J. S., 2006, “Maximizing Band Gaps in Plate Structures,” Struct. Multidiscip. Optim., 32(4), pp. 263–275. [CrossRef]
Diaz, A. R., Haddow, A. G., and Ma, L., 2005, “Design of Band-Gap Grid Structures,” Struct. Multidiscip. Optim., 29(6), pp. 418–431. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Jacquot, R. G., and Foster, J. E., 1977, “Optimal Cantilever Dynamic Vibration Absorbers,” ASME J. Eng. Ind., 99(1), pp. 138–141. [CrossRef]
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. [CrossRef]
Brennan, M. J., 1997, “Characteristics of a Wideband Vibration Neutralizer,” Noise Control Eng. J., 45(5), pp. 201–207. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Graff, K. F., 1975, Wave Motion in Elastic Solids, Oxford University, London.
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. [CrossRef]
Lee, U., 2009, Spectral Element Method in Structural Dynamics, Wiley, Singapore.
Brillouin, L., 1946, Wave Propagation in Periodic Structures, Dover, New York.
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. [CrossRef]
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. [CrossRef]

Figures

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Fig. 2

Schematic diagram of the separated beam-like resonator

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Fig. 1

(a) Schematic diagram of the designed LR beam structure. (b) Simplified physical model.

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Fig. 3

The approximate lumped system of the cantilever beam segment shown in Fig. 2

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Fig. 4

One unit cell of the LR beam shown in Fig. 1(b)

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Fig. 5

(a) Schematic diagram of a finite LR beam. (b) Numbering of the spectral beam elements and nodes.

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Fig. 6

The driving-point dynamic stiffness of the beam-like resonator tuned at fr = 777 Hz

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Fig. 7

(a) Schematic and (b) photograph of the experimental setup

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Fig. 8

Complex band structure for the LR beam with resonators tuned at fr = 869 Hz. (a) 200–2000 Hz; (b) 900–960 Hz.

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Fig. 11

Attenuation constants for the LR beam with resonators tuned at fr = 1269 Hz

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Fig. 9

The smaller attenuation constant for the LR beam with resonators tuned at fr = 869 Hz

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Fig. 10

(a) Evolution of band gap behavior with changing total length of the beam-like resonator: 2l. (b) Closer view of the area denoted by the pane in (a).

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Fig. 13

Configuration of the finite LR beam model adopted for the finite element simulation

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Fig. 14

Comparison between the analytically predicted (SEM) and numerically simulated (FEM) vibration transmittance FRF of the finite LR beam with resonators tuned at fr = 869 Hz

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Fig. 15

Steady-state vibration profiles (in x–z plane) at the frequencies denoted by the points (a) A, (b) B, (c) C, and (d) D in Fig. 14

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Fig. 16

Influence of (a) the structural length and (b) the global resonator location on the vibration transmittance of finite LR beams with resonators tuned at fr = 869 Hz

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Fig. 12

(a) Analytically predicted band edge frequency curves. (b) Closer view of the area denoted by the pane in (a).

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Fig. 18

(a) Vibration transmittance FRF and (b) smaller attenuation constant for the LR beam with resonators tuned at fr = 777 Hz

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Fig. 17

Effects of (a) the host-beam damping and (b) the resonator damping on the band gap behavior (upper panel) and vibration transmittance (lower panel) of LR beams with resonators tuned at fr = 869 Hz

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Fig. 19

(a) Vibration transmittance FRF and (b) smaller attenuation constant for the LR beam with resonators tuned at fr = 869 Hz

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Fig. 20

(a) Vibration transmittance FRF and (b) smaller attenuation constant for the LR beam with resonators tuned at fr = 978 Hz

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Fig. 21

(a) Vibration transmittance FRF and (b) smaller attenuation constant for the LR beam with resonators tuned at fr = 1110 Hz

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Fig. 22

(a) Vibration transmittance FRF and (b) smaller attenuation constant for the LR beam with resonators tuned at fr = 1269 Hz

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Fig. 23

Comparison of vibration transmittances for two cases of placement of the load

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Fig. 24

Numerical and experimental vibration transmittance FRF for the modified version (aL = 105 mm) of the LR beam with resonators tuned at fr = 869 Hz

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