WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Chapter 2
The bandgap phenomenon
Inorder tounderstand complex phenomena itisalways useful tolookatthesimplest
possible structure that displays the behavior of interest. This obvious observation
fully applies to the phenomenon of the formation of gaps in the band structure
for periodic materials. A system of simple masses connected by linear springs is
particularlyuseful becauseitproducessimpleequationsoftenamenabletoanalytical
predictions. Additionally, visualization of the dynamic behavior is straightforward
and this aids immensely in the process of grasping the mechanisms involved.
All theoretical results presented in this chapter are obtained using simple massspring
models. This chapter intends to provide a fundamental understanding of the
bandgap phenomenon and pinpoint the factors that have significant impact on the
performance of engineering structures that integrate bandgap materials.
Thesis papers [1]–[3]
Paper [1] presents an analysis of the interaction between bandgaps and forced vibrations
for 1D and 2D mass-spring systems. The focus is on these issues: the
effects of the finite dimensions of structures created from a bandgap material, the
effect of non-idealities such as damping and disorder and the possibility of creating
waveguides in bandgap structures. These effects are analyzed for different types
of bandgap materials. One of these offers the possibility of creating low-frequency
bandgaps.
Paper[2]demonstrates experimentally thebandgapphenomenonforlongitudinal
vibrations in an elastic rod composed of sections of aluminum and PMMA (plastic).
The experimental results are compared to predictions based on the mass-spring
model studied in paper [1]. Paper [2] contains preliminary optimization results for
bandgap structures which are relevant to the results presented in Chapter 3.
Paper[3]analyzesthebandgapbehaviorandwave transmissioninafinitenonlinear
structure using analytical and numerical tools. It focuses on a one-dimensional
mass-springsystemwithnonlinearresonatorsattachedwithrespecttolow-frequency
bandgaps. Special attention is given to the effect of the nonlinearities on the wave
transmission through the chain. Furthermore, the paper includes a discussion of the
possibilities of utilizing a non-uniform distribution of nonlinearities to reduce the
wave transmission.
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