Existing band gap analysis is mostly focused on the binary structure, while the researches on the quaternary layered periodic structure are still lacking. In this paper, the unidimensional lumped-mass method in the phonic crystal theory is firstly improved so that the material viscoelasticity can be taken into consideration. Then, the binary layered periodic structure is converted into a quaternary one and band gaps appear at low frequency range. Finally, the effects of density, elastic modulus, damping ratio, and the thickness of single material on the first band gap of the quaternary layered periodic structure are analyzed after the algorithm is promoted. The research findings show that effects of density, elastic modulus, and thickness of materials on the first band gap are considerable but those of damping ratio are not so distinct. This research provides theoretical bases for band gap design of the quaternary layered periodic structure. 1. Introduction In recent years, the researches on energy band features of periodic structure constructed artificially based on the phononic crystal theory have drawn more and more attention. This is because artificially constructed periodic structure is characterized by bandpass and bandstop of classical waves. In the acoustics, the gaps between energy bands are called phononic band gaps and the corresponding periodic structure is called phononic crystal. As early as in the 1980s, Achenbach and Kitahara [1, 2] studied the propagation of elastic waves in periodic medium with spherical cavity. Sigalas and Economou [3] first theoretically demonstrated that band gaps appeared in periodic lattice structure formed by embedding solid spherical materials into the matrix. Afterwards, the researchers [4–6] studied the influence of material properties on phononic band gaps and found that the main reason why phononic band gaps occurred lied in periodic variation of material properties like the density and elastic modulus. The layered periodic structure, formed by two or more types of materials that are alternatively arranged, belongs to unidimensional phononic crystal and has attracted wide attention of domestic and overseas scholars. Esquivel-Sirvent and Cocoletzi [7] deduced the reflection coefficient and chromatic dispersion relation for the propagation of elastic waves in layered periodic structure. Hussein et al. [8, 9] studied the layered periodic structure constructed with two types of materials and obtained the band gap features in the variation of material distribution in each period. Meanwhile, they observed the
References
[1]
J. D. Achenbach and M. Kitahara, “Reflection and transmission of an obliquely incident wave by an array of spherical cavities ?” Journal of the Acoustical Society of America, vol. 80, pp. 1209–1214, 1986.
[2]
J. D. Achenbach and M. Kitahara, “Harmonic waves in a solid with a periodic distribution of spherical cavities,” Journal of the Acoustical Society of America, vol. 81, pp. 595–598, 1987.
[3]
M. M. Sigalas and E. N. Economou, “Elastic and acoustic wave band structure,” Journal of Sound and Vibration, vol. 158, no. 2, pp. 377–382, 1992.
[4]
M. Sigalas and E. N. Economou, “Band structure of elastic waves in two dimensional systems,” Solid State Communications, vol. 86, no. 3, pp. 141–143, 1993.
[5]
M. S. Kushwaha, P. Halevi, G. Martínez, L. Dobrzynski, and B. Djafari-Rouhani, “Theory of acoustic band structure of periodic elastic composites,” Physical Review B, vol. 49, no. 4, pp. 2313–2322, 1994.
[6]
C.-S. Kee, J.-E. Kim, H. Y. Park, K. J. Chang, and H. Lim, “Essential role of impedance in the formation of acoustic band gaps,” Journal of Applied Physics, vol. 87, no. 4, pp. 1593–1596, 2000.
[7]
R. Esquivel-Sirvent and G. H. Cocoletzi, “Band structure for the propagation of elastic waves in superlattices,” Journal of the Acoustical Society of America, vol. 95, no. 1, pp. 86–90, 1994.
[8]
M. I. Hussein, G. M. Hulbert, and R. A. Scott, “Dispersive elastodynamics of 1D banded materials and structures: analysis,” Journal of Sound and Vibration, vol. 289, no. 4-5, pp. 779–806, 2006.
[9]
M. I. Hussein, G. M. Hulbert, and R. A. Scott, “Dispersive elastodynamics of 1D banded materials and structures: design,” Journal of Sound and Vibration, vol. 307, no. 3–5, pp. 865–893, 2007.
[10]
Y. P. Zhao and P. J. Wei, “The band gap of 1D viscoelastic phononic crystal,” Computational Materials Science, vol. 46, no. 3, pp. 603–606, 2009.
[11]
M.-L. Wu, L.-Y. Wu, W.-P. Yang, and L.-W. Chen, “Elastic wave band gaps of one-dimensional phononic crystals with functionally graded materials,” Smart Materials and Structures, vol. 18, no. 11, Article ID 115013, 2009.
[12]
P. D. Sesion Jr., E. L. Albuquerque, C. Chesman, and V. N. Freire, “Acoustic phonon transmission spectra in piezoelectric AlN/GaN Fibonacci phononic crystals,” European Physical Journal B, vol. 58, no. 4, pp. 379–387, 2007.
[13]
F.-M. Li, Y.-Z. Wang, B. Fang, and Y.-S. Wang, “Propagation and localization of two-dimensional in-plane elastic waves in randomly disordered layered piezoelectric phononic crystals,” International Journal of Solids and Structures, vol. 44, no. 22-23, pp. 7444–7456, 2007.
[14]
F.-M. Li, M.-Q. Xu, and Y.-S. Wang, “Frequency-dependent localization length of SH-wave in randomly disordered piezoelectric phononic crystals,” Solid State Communications, vol. 141, no. 5, pp. 296–301, 2007.
[15]
Y.-Z. Wang, F.-M. Li, W.-H. Huang, and Y.-S. Wang, “The propagation and localization of Rayleigh waves in disordered piezoelectric phononic crystals,” Journal of the Mechanics and Physics of Solids, vol. 56, no. 4, pp. 1578–1590, 2008.
[16]
Y.-Z. Wang, F.-M. Li, K. Kishimoto, Y.-S. Wang, and W.-H. Huang, “Wave localization in randomly disordered layered three-component phononic crystals with thermal effects,” Archive of Applied Mechanics, vol. 80, no. 6, pp. 629–640, 2010.
[17]
H. Policarpo, M. M. Neves, and A. M. R. Ribeiro, “Dynamical response of a multi-laminated periodic bar: analytical, numerical and experimental study,” Shock and Vibration, vol. 17, no. 4-5, pp. 521–535, 2010.
[18]
J. S. Jensen, “Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures,” Journal of Sound and Vibration, vol. 266, no. 5, pp. 1053–1078, 2003.
[19]
G. Wang, J. Wen, Y. Liu, and X. Wen, “Lumped-mass method for the study of band structure in two-dimensional phononic crystals,” Physical Review B, vol. 69, no. 18, Article ID 184302, pp. 1–184302, 2004.
[20]
D. Richards and D. J. Pines, “Passive reduction of gear mesh vibration using a periodic drive shaft,” Journal of Sound and Vibration, vol. 264, no. 2, pp. 317–342, 2003.
