WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Phononic band-gap optimization
(aI) (b) (c)
1 - *_ - m - -
O- ----- -
-pu materia 1 -- - -
i
20 r Xr
20 2
I , i
IL'
I I a
r * * * ,__ z
ma -20 V'
2 -60 -
-80O
periodic
periodic with damping
pure material 1
pure material 2
I full gap
t100 L__
0 20
I
40
1
60
I I
80 100
frequency, f (kHz)
I
II
-
1007
Figure 4. Response of a 12 cm x 12 cm square structure subjected to periodic loading on the left
boundary, (a) pure material 1, (b) pure material 2, and (c) a periodic structure of materials 1
and 2; (d) response calculated as the average amplitude on the right boundary for the structure
of pure materials 1 and 2 without damping, and for the periodic structure with and without
mass-proportional damping. High-contrast case.
and second bands (from 38 to 46 kHz, corresponding to a relative band-gap size of
A f/fo = 0.20). Furthermore, there are two partial band gaps (for modes propagat-
ing in the horizontal direction for the frequency ranges 28-46 kHz and 70-76 kHz
(indicated by cross-hatched regions in the figure). Note that the calculations in this
example are based on an extremely coarse discretization (10 x 10 elements). The
coarse discretization is chosen in order to be able to compare the results with results
for the finite-dimensional structures that cannot be modelled with an extremely fine
grid due to computing-time limitations.
(b) The structural problem
For the structural problem with external harmonic loading, we assume a harmonic
wave solution described as
u = eit, (2.10)
where Q is the driving frequency and ut is the amplitude. Substituting equation (2.10)
into equations (2.4), (2.5) or equation (2.6), depending on whether in-plane or out-
of-plane waves are considered, dropping the hat, and converting to FEM notation
Phil. Trans. R. Soc. Lond. A (2003)