WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

14 Chapter 2 The bandgap phenomenon
Amplitude (last mass)
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a) Frequency b) Frequency
Figure 2.8 Amplitude of last mass (relative to the first mass) for a one-dimensional
mass-spring structure with 2000 attached oscillators. a) Linear oscillators, b) Nonlinear
oscillators. Solid lines: numerical simulation, dotted lines: analytical predictions of upper
and lower amplitude bounds. From paper [3].
quency range and the behavior is much more complex due to the varying bandgap
behavior along the structure and the nonlinear wave interaction. However, it can be
observed that theamplitude reduction issmaller thanforthe linear case, but thefrequency
range with a significant reduction of the amplitude is slightly broader. The
figures alsoillustrate upper andlower boundsfortheresponse predicted analytically.
Theperspectivesforusinganon-homogeneousdistributionofnonlinearitiesalong
the chain is illustrated in Fig. 2.9 for the case of 400 attached oscillators. The solid
line represents the wave transmission when the nonlinear parameter is increased
along the chain according to a prescribed exponential function. This is done to compensate
for the reduced wave amplitude and the resulting decrease in the bandgap
frequency. The transmission is also shown with three different constant values of
the nonlinear parameter. These values correspond to the minimum, maximum and
the average value of the exponential parameter variation. It is seen that the nonhomogeneous
distribution opens up the possibilities of further wave propagation
suppression. These findings have prompted the work on optimization of local oscillator
parameters as presented in Section 5.2.
Relations to recent work
The research on bandgaps has continued with increasing intensity and a large number
of papers have appeared since the publication of paper [1].
Paper [1] contributed to the basic understanding of the bandgap phenomenon by
studying simple mass-spring systems. Other researchers have continued this work
up until the present date. An incomplete compilation of recent theoretical papers
on mass-spring systems include the work by Avila and Reyes (2008) on phonon
propagation through 1D atomic structures, by Zalipaev et al. (2008) on waves in
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