Research Papers

Analysis and Interpretation of Longitudinal Waves in Periodic Multiphase Rods Using the Method of Reverberation-Ray Matrix Combined With the Floquet-Bloch Theorem

[+] Author and Article Information
Y. Q. Guo

State Key Laboratory of Turbulence and
Complex Systems;
College of Engineering,
Peking University,
Beijing 100871, China;
Key Laboratory of Mechanics on Disaster and
Environment in Western China,
Ministry of Education,
School of Civil Engineering and Mechanics,
Lanzhou University,
Lanzhou 730000, China
e-mail: guoyq@lzu.edu.cn

D. N. Fang

State Key Laboratory of Turbulence and
Complex Systems,
College of Engineering,
Peking University,
Beijing 100871, China

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 1, 2012; final manuscript received September 4, 2013; published online October 23, 2013. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 136(1), 011006 (Oct 23, 2013) (13 pages) Paper No: VIB-12-1336; doi: 10.1115/1.4025438 History: Received December 01, 2012; Revised September 04, 2013

The method of reverberation-ray matrix (MRRM) combined with the Floquet–Bloch theorem, which serves as an alternative method, is presented for accurately analyzing longitudinal waves in general periodic multiphase rods. Closed-form dispersion relation of periodic quaternary rods is derived. Based on this relation, the functions of constituent-rod parameters in the formation of longitudinal-wave band structures are analytically revealed. Numerical examples validate the proposed method and indicate the characteristics/applications of all kinds of dispersion curves that include the frequency-wave number spectra, the frequency-wavelength spectra, the frequency-phase velocity spectra, the wave number-phase velocity spectra and the wavelength-phase velocity spectra. The effect of unit-cell layout on the frequency band properties and the functions of constituent-rod parameters in the band structure formation are also illustrated numerically. The analysis and interpretation of longitudinal waves in periodic multiphase rods given in this paper will push forward the design of periodic structures for longitudinal wave filtering/guiding and vibration isolation/control applications.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Brillouin, L., 1953, Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, 2nd ed., Dover, New York.
Mead, D. J., 1996, “Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton, 1964-1995,” J. Sound Vib., 190, pp. 495–524. [CrossRef]
Sen Gupta, G., 1980, “Vibration of Periodic Structures,” Shock Vib. Dig., 12, pp. 17–31. [CrossRef]
Li, D., and Benaroya, H., 1992, “Dynamics of Periodic and Near-Periodic Structures,” ASME Appl. Mech. Rev., 45, pp. 447–459. [CrossRef]
Mester, S. S., and Benaroya, H., 1995, “Periodic and Near-Periodic Structures,” Shock Vib., 2(1), pp. 69–95. [CrossRef]
Elachi, C., 1976, “Waves in Active and Passive Periodic Structures: A Review,” Proc. IEEE, 64, pp. 1666–1698. [CrossRef]
Johnson, S. G., and Joannopoulos, J. D., 2002, Photonic Crystal: The Road From Theory to Practice, Kluwer, Boston.
Kushwaha, M. S., 1996, “Classical Band Structure of Periodic Elastic Composites,” Int. J. Mod. Phys. B, 10, pp. 977–1094. [CrossRef]
Miyashita, T., 2005, “Sonic Crystals and Sonic Wave-Guides,” Meas. Sci. Technol., 16, pp. R47–R63. [CrossRef]
Lu, M.-H., Feng, L., and Chen, Y.-F., 2009, “Phononic Crystals and Acoustic Metamaterials,” Mater. Today, 12(12), pp. 34–42. [CrossRef]
Wen, J. H., Wang, G., Yu, D. L., Zhao, H. G., Liu, Y. Z., and Wen, X. S., 2008, “Study on the Vibration Band Gap and Vibration Attenuation Property of Phononic Crystals,” Sci. China Ser. E-Technol. Sci., 51, pp. 85–99. [CrossRef]
Sigalas, M. M., and Economou, E. N., 1992, “Elastic and Acoustic Wave Band Structure,” J. Sound Vib., 158(2), pp. 377–382. [CrossRef]
Sigalas, M., Kushwaha, M. S., Economou, E. N., Kafesaki, M., Psarobas, I. E., and Steurer, W., 2005, “Classical Vibrational Modes in Phononic Lattices: Theory and Experiment,” Z. Kristallogr., 220, pp. 765–809. [CrossRef]
Sigmund, O., and Jensen, J. S., 2003, “Systematic Design of Phononic Band-Gap Materials and Structures by Topology Optimization,” Philos. Trans. R. Soc. London, Ser. A, 361, pp. 1001–1019. [CrossRef]
Jensen, J. S., and Pedersen, N. L., 2006, “On Maximal Eigenfrequency Separation in Two-Material Structures: The 1D and 2D Scalar Cases,” J. Sound Vib., 289, pp. 967–986. [CrossRef]
Hussein, M. I., Hamza, K., Hulbert, G. M., Scott, R. A., and Saitou, K., 2006, “Multiobjective Evolutionary Optimization of Periodic Layered Materials for Desired Wave Dispersion Characteristics,” Struct. Multidisc. Optim., 31, pp. 60–75. [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, pp. 865–893. [CrossRef]
Asiri, S., Baz, A., and Pines, D., 2005, “Periodic Struts for Gearbox Support System,” J. Vib. Control, 11, pp. 709–721. [CrossRef]
Asiri, S., 2005, “Vibration Isolation of Automotive Vehicle Engine Using Periodic Mounting Systems,” Proc. SPIE, 5760, pp. 526–537. [CrossRef]
Policarpo, H., Neves, M. M., and Ribeiro, A. M. R., 2010, “Dynamical Response of a Multi-Laminated Periodic Bar: Analytical, Numerical and Experimental Study,” Shock Vib., 17, pp. 521–535. [CrossRef]
Shen, H. J., Wen, J. H., Yu, D. L., and Wen, X. S., 2009, “The Vibrational Properties of a Periodic Composite Pipe in 3D Space,” J. Sound Vib., 328, pp. 57–70. [CrossRef]
Cremer, L., Heckl, M., and Petersson, B. A. T., 2005, Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies, 3rd ed., Springer, Berlin, pp. 380–385.
Ravindra, B., 1999, “Love-Theoretical Analysis of Periodic System of Rods,” J. Acoust. Soc. Am., 106(2), pp. 1183–1186. [CrossRef] [PubMed]
Ohlrich, M., 1986, “Forced Vibration and Wave Propagation in Mono-Coupled Periodic Structures,” J. Sound Vib., 107, pp. 411–434. [CrossRef]
Wang, G., Wen, X. S., Wen, J. H., and Liu, Y. Z., 2006, “Quasi-One-Dimensional Periodic Structure With Locally Resonant Band Gap,” ASME J. Appl. Mech., 73, pp. 167–170. [CrossRef]
Pai, P. F., 2010, “Metamaterial-Based Broadband Elastic Wave Absorber,” J. Intell. Mater. Syst. Struct., 21, pp. 517–528. [CrossRef]
Xiao, Y., Wen, J. H., and Wen, X. S., 2012, “Longitudinal Wave Band Gaps in Metamaterial-Based Elastic Rods Containing Multi-Degree-of-Freedom Resonators,” New J. Phys., 14, p. 033042. [CrossRef]
Yeh, J.-Y., and Chen, L.-W., 2006, “Wave Propagations of a Periodic Sandwich Beam by FEM and the Transfer Matrix Method,” Compos. Struct., 73, pp. 53–60. [CrossRef]
Li, D., and Benaroya, H., 1994, “Waves, Normal Modes and Frequencies of Periodic and Near-Periodic Rods-I,” Wave Motion, 20, pp. 315–338. [CrossRef]
Vakakis, A. F., Raheb, M. E., and Cetinkaya, C., 1994, “Free and Forced Dynamics of a Class of Periodic Elastic Systems,” J. Sound Vib., 172(1), pp. 23–46. [CrossRef]
Andrianov, I. V., Bolshakov, V. I., Danishevs'kyy, V. V., and Weichert, D., 2008, “Higher Order Asymptotic Homogenization and Wave Propagation in Periodic Composite Materials,” Proc. R. Soc. A, 464, pp. 1181–1201. [CrossRef]
Keane, A. J., and Price, W. G., 1989, “On the Vibrations of Mono-Coupled Periodic and Near-Periodic Structures,” J. Sound Vib., 128(3), pp. 423–450. [CrossRef]
Morales, A., Flores, J., Gutierrez, L., and Mendez-Sanchez, R. A., 2002, “Compressional and Torsional Wave Amplitudes in Rods With Periodic Structures,” J. Acoust. Soc. Am., 112(5), pp. 1961–1967. [CrossRef] [PubMed]
Guo, Y. Q., and Fang, D. N., 2011, “Formation of Longitudinal Wave Band Structures in One-Dimensional Phononic Crystals,” J. Appl. Phys., 109(7), p. 073515. [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, pp. 779–806. [CrossRef]
Shen, M. R., and Cao, W. W., 2000, “Acoustic Bandgap Formation in a Periodic Structure With Multilayer Unit Cells,” J. Phys. D-Appl. Phys., 33, pp. 1150–1154. [CrossRef]
Mead, D. J., 1971, “Vibration Response and Wave Propagation in Periodic Structures,” ASME J. Eng. Ind., 93, pp. 783–792. [CrossRef]
Sen Gupta, G., 1971, “Natural Frequencies of Periodic Skin-Stringer Structures Using a Wave Approach,” J. Sound Vib., 16(4), pp. 567–580. [CrossRef]
Mead, D. J., 1970, “Free Wave Propagation in Periodically Supported, Infinite Beams,” J. Sound Vib., 11(2), pp. 181–197. [CrossRef]
Howard, S. M., and Pao, Y. H., 1998, “Analysis and Experiments on Stress Waves in Planar Trusses,” ASCE J. Eng. Mech., 124, pp. 884–891. [CrossRef]
Pao, Y. H., Keh, D. C., and Howard, S. M., 1999, “Dynamic Response and Wave Propagation in Plane Trusses and Frames,” AIAA J., 37, pp. 594–603. [CrossRef]
Guo, Y. Q., 2008, “The Method of Reverberation-Ray Matrix and Its Applications” (in Chinese), Ph.D. thesis, Zhejiang University, Hangzhou, China.
Pao, Y. H., and Chen, W. Q., 2009, “Elastodynamic Theory of Framed Structures and Reverberation-Ray Matrix Analysis,” Acta Mech., 204, pp. 61–79. [CrossRef]
Mead, D. J., 1973, “A General Theory of Harmonic Wave Propagation in Linear Periodic Systems With Multiple Coupling,” J. Sound Vib., 27(2), pp. 235–260. [CrossRef]
Graff, K. F., 1975, Wave Motion in Elastic Solids, Ohio State University Press, Columbus, OH, pp. 75–125.
Esquivel-Sirvent, R., and Cocoletzi, G. H., 1994, “Band Structure for the Propagation of Elastic Waves in Superlattices,” J. Acoust. Soc. Am., 95(1), pp. 86–90. [CrossRef]


Grahic Jump Location
Fig. 3

Typical parts of the unit cell: the coordinates and the physical variables

Grahic Jump Location
Fig. 2

The displacements/forces at the ends of the current and the adjacent unit cells

Grahic Jump Location
Fig. 1

The general infinite periodic multiphase rod and the selected unit cell

Grahic Jump Location
Fig. 5

The frequency-wave number spectra of longitudinal waves in the exemplified periodic hexadecimal rod

Grahic Jump Location
Fig. 6

The frequency-wavelength spectra of longitudinal waves in the exemplified periodic hexadecimal rod

Grahic Jump Location
Fig. 7

The frequency-phase velocity spectra of longitudinal waves in the exemplified periodic hexadecimal rod

Grahic Jump Location
Fig. 10

Effect of the cross-sectional area of the constituent rod on the band structure formation

Grahic Jump Location
Fig. 4

The obtained first several wave number-frequency spectra of characteristic longitudinal waves in the periodic binary-I and binary-II rods and their comparisons with the existing counterparts

Grahic Jump Location
Fig. 12

Effect of the mass density of the constituent rod on the band structure formation

Grahic Jump Location
Fig. 11

Effect of the length of the constituent rod on the band structure formation



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In