In-plane Band Gaps in a Periodic Plate with Piezoelectric Patches

Document Type: Research Paper


School of Civil Engineering, Beijing Jiaotong University


A plate periodically bonded with piezoelectric patches on its surfaces is considered. The differential quadrature element method is used to solve the wave motion equation for the two-dimensional periodic structure. The method is very simple and easy to implement. Based on the method, band structures for in-plane wave propagating in the periodic piezoelectric plate are studied, from which the frequency band gap is then obtained. Parametric studies are also performed to highlight geometrical and physical parameters on the band gaps. It is found that the thickness of the piezoelectric patches have no effect on the upper bound frequency of the band gap. Physical mechanism is explained for the phenomena. Dynamic simulations are finally conducted to show how the band gap works for a finite quasi-periodic plate. Numerical results show that the vibration in periodic plates can be dramatically attenuated when the exciting frequency falls into the band gap.                                                  


[1] Sigalas M., Kushwaha M.S., Economou E.N., Kafesaki M., Psarobas I.E., Steurer W., 2005, Classical vibrational modes in phononic lattices: theory and experiment, Zeitschrift für Kristallographie 220:765-809.
[2] Baz A., 2001, Active control of periodic structures, Journal of Vibration and Acoustic 123: 472-479.
[3] Kushwaha M.S., Halevi P., Dobrzynski L., Djafari-Rouhani B., 1993, Acoustic band structure of periodic elastic composites, Physical Review Letters 71(13): 2022-2025.
[4] Kushwaha M.S., Halevi P., 1994, Band gap engineering in periodic elastic composites, Applied Physics Letters 64(9): 1085-1087.
[5] Liu Z., Zhang X., Mao Y., Zhu Y.Y., Yang Z., Chan C.T., Sheng P., 2000, Locally resonant sonic materials, Science 289(5485): 1734-1736.
[6] Jia G., Shi Z., 2010, A new seismic isolation system and its feasibility study, Earthquake Engineering and Engineering Vibration 9(1): 75-82.
[7] Bao J., Shi Z.F., Xiang H.J., 2012, Dynamic responses of a structure with periodic foundations, Journal of Engineering Mechanics-ASCE 138(7): 761-769.
[8] Xiang H.J., Shi Z.F., Wang S.J., Mo Y.L., 2012, Periodic materials-based vibration attenuation in layered foundations: experimental validation, Smart Materials and Structures 21(11): 112003.
[9] Xiong C., Shi Z.F., Xiang H.J., 2012, Attenuation of building vibration using periodic foundations, Advances in Structural Engineering 15(8): 1375-1388.
[10] Khelif A., Djafari-Rouhani B., Vasseur J.O., Deymier P.A., Lambin P., Dobrzynski L., 2002, Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal, Physical Review B 65(17): 174308.
[11] Khelif A., Aoubiza B., Mohammadi S., Adibi A., Laude V., 2006, Complete band gaps in two-dimensional phononic crystal slabs, Physical Review E 74(4): 046610.
[12] Thorp O., Ruzzene M., Baz A., 2001, Attenuation and localization of wave propagation in rods with periodic shunted piezoelectric patches, Smart Materials and Structures 10: 979.
[13] Wang Y.Z., Li F.M., Kishimoto K., Wang Y.S., Huang W.H., 2009, Wave band gaps in three-dimensional periodic piezoelectric structures, Mechanics Research Communications 36(4): 461-468.
[14] Wu T.T., Hsu Z.C., Huang Z.G., 2005, Band gaps and the electromechanical coupling coefficient of a surface acoustic wave in a two-dimensional piezoelectric phononic crystal, Physical Review B 71(6): 064303.
[15] Zou X.Y., Chen Q., Liang B., Cheng J.C., 2008, Control of the elastic wave bandgaps in two-dimensional piezoelectric periodic structures, Smart Materials and Structures 17: 015008.
[16] Xiang H.J., Shi Z.F., 2009, Analysis of flexural vibration band gaps in periodic beams using differential quadrature method, Computers & Structures 87(23-24): 1559-1566.
[17] Xiang H.J., Shi Z.F., 2011, Vibration attenuation in periodic composite Timoshenko beams on Pasternak foundation, Structural Engineering and Mechanics 40(3): 373-392.
[18] Kittel C., 2005, Introduction to Solid State Physics, John Wiley & Son, New York, 8th Edition.
[19] Chen C.N., 2004, Extended GDQ and related discrete element analysis methods for transient analyses of continuum mechanics problems, Computers & Mathematics with Applications 47(1): 91-99.
[20] Chen C.N., 2008, DQEM analysis of out-of-plane vibration of nonprismatic curved beam structures considering the effect of shear deformation, Advances in Engineering Software 39(6): 466-472.
[21] Malekzadeh P., Karami G., Farid M., 2004, A semi-analytical DQEM for free vibration analysis of thick plates with two opposite edges simply supported, Computer Methods in Applied Mechanics and Engineering 193(45-47): 4781-4796.
[22] Liu F.L., Liew K.M., 1999, Differential quadrature element method: a new approach for free vibration analysis of polar Mindlin plates having discontinuities, Computer Methods in Applied Mechanics and Engineering 179(3): 407-423.
[23] Eisinberg A., Fedele G., 2005, Vandermonde systems on gauss-lobatto chebyshev nodes, Applied Mathematics and Computation 170(1): 633-647.
[24] Shu C., 2000, Differential Quadrature: And Its Application in Engineering, Springer, London.
[25] Shu C., Richards B.E., 1992, Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids 15(7): 791-798.
[26] Xiang H.J., Yang J., 2008, Free and forced vibration of a laminated FGM timoshenko beam of variable thickness under heat conduction, Composites Part B-Engineering 39(2): 292-303.
[27] Yang J., Xiang H.J., 2007, Thermo-electro-mechanical characteristics of functionally graded piezoelectric actuators, Smart Materials and Structures 16(3): 784-797.
[28] Wen X.S., Wen J.H., Yu D.L., Wang G., Liu Y.Z., Han X.Y., 2009, Phononic Crystals, Ational Defense Industry Press, Beijing.
[29] Åberg M., Gudmundson P., 1997, The usage of standard finite element codes for computation of dispersion relations in materials with periodic microstructure, The Journal of the Acoustical Society of America 102(4): 2007-2013.
[30] Yaman Y., Çalışkan T., Nalbantoğlu V., Prasad E., Waechter D., 2002, Active Vibration Control of a Smart Plate, ICAS2002, Toronto, Canada.