WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

4. Conclusions
In order to use phononic band gap materials in mechanical structures that utilize the selective
wave blocking abilities of these materials, the effects associated with the boundaries, damping,
and imperfections need to be taken into account.
In this work the vibrational response of 1-D and 2-D mass–spring structures subjected to
periodic loading has been investigated. The focus has been put on analyzing structures with a
periodic micro-structure (unit cells) with gaps in the band structure for the corresponding infinite
periodic lattice. Two examples were used to demonstrate the effect on the response of boundaries,
viscous damping, and imperfections—a 1-D structure acting as a wave filter, and wave guiding in
2-D structures.
It was shown how the response in the band gap frequency range depends on the number of unit
cells in the structure. In 1-D and for one of the two types of unit cells analyzed in 2-D, it was seen
how the response in the band gap is insensitive to moderate amounts of viscous damping, whereas
for the other 2-D unit cell analyzed, the band gap effect almost disappears with strong damping
added. In the 1-D example it was shown also that the response in the band gap is insensitive to
small imperfections in the periodic structure. It was demonstrated in both the 1-D and 2-D
example, that the free boundaries cause local resonances that affect the response in the band gap.
Two 2-D wave guides were analyzed, created by replacing the periodic structure with a
homogeneous structure in a straight and a 901 bent path. It was shown how the vibrational
response is confined to these paths in the band gap frequency ranges.
Further work deals with the formulation and solution of optimization problems that can be
used to design of applications of phononic band gap materials and structures.
Acknowledgements
This work was supported by the Danish Research Council through the project ‘‘Phononic
bandgap materials: Analysis and optimization of wavetransmission in periodic materials’’. The
author would like to thank Jon Juel Thomsen, Ole Sigmund, and Martin P. Bends^e for valuable
comments and suggestions.
References
ARTICLE IN PRESS
J.S. Jensen / Journal of Sound and Vibration 266 (2003) 1053–1078 1077
[1] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals, Molding the Flow of Light, Princeton University
Press, Princeton, NJ, 1995.
[2] E. Yablonovitch, Photonic crystals: semiconductors of light, Scientific American 285 (2001) 34–41.
[3] Lord Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation of waves
through a medium endowed with a periodic structure, Philosophical Magazine 24 (1887) 145–159.
[4] L. Brillouin, Wave Propagation in Periodic Structures, Dover Publications Inc, New York, 1953.
[5] C. Elachi, Waves in active and passive periodic structures: a review, Proceedings of the IEEE 64 (1976) 1666–1698.
[6] D.J. Mead, Wave propagation in continuous periodic structures: research contributions from Southampton,
1964–1995, Journal of Sound and Vibration 190 (1996) 495–524.
[7] S. Parmley, T. Zobrist, T. Clough, A. Perez-miller, M. Makela, R. Yu, Phononic band structure in a mass chain,
Applied Physics Letters 67 (1995) 777–779.