WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
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898 S. Halkjær, O. Sigmund and J. S. Jensen<br />
Alu<br />
PMMA<br />
f [kHz]<br />
100<br />
80<br />
60<br />
40<br />
20<br />
1 2 3 4 5 6 7 8<br />
[cm]<br />
0<br />
0 0.5 1 1.5<br />
k<br />
B<br />
2 2.5 3<br />
Fig. 1. Final design corresponding to maximum relative band gap between<br />
first and second band (thick lines) for longitudinal waves. Full<br />
lines: Longitudinal waves. Dashed lines: Bending waves.<br />
Table 2. Extreme frequency band values for longitudinal waves in<br />
Figure 1. Unit is kHz.<br />
k B ¼ 0 k B ¼ p<br />
f1 0 10.367<br />
f2 40.798 30.452<br />
f3 40.844 51.191<br />
f4 81.603 71.283<br />
design. It should be noted, that if another base cell length<br />
was chosen, no qualitative change would be observed and<br />
the ratio of the two material lengths would be the same.<br />
With this said, PMMA occupies approximately a fraction<br />
of 0.29 of the base cell length. That is, the larger part of<br />
the base cell is made up of the heavy and stiff material<br />
while a smaller part is occupied by the light and soft material.<br />
Figure 1 (bottom) also shows the lowest frequency<br />
bands for both longitudinal and bending waves, the latter<br />
just to indicate their position. The extreme edge values of<br />
the bands are listed in Table 2. The first three relative<br />
band gap sizes are listed in Table 6 (second column). The<br />
symbols in the first column denote band gap number and<br />
wave type considered, e.g. F L 12<br />
denotes the band gap be-<br />
tween the first and second frequency band and L indicates,<br />
that longitudinal waves are considered.<br />
As a second example the relative band gap between the<br />
third and fourth band for bending waves has been maximized.<br />
The resulting design and frequency bands are<br />
shown in Fig. 2. In this case, increasing or decreasing the<br />
base cell length results in an increasing relative band gap<br />
size mainly due to vertical displacements of the frequency<br />
bands. This effect has been excluded by choosing the<br />
same constant base cell length as in the previous case<br />
d ¼ 8:70 cm. This design problem appears to have many<br />
local minima, some of which result in designs consisting<br />
of mixtures of the two materials (i.e. z e 6¼ 0; 1), depending<br />
on the starting point. The design shown in Fig. 2 has the<br />
largest relative band gap of several trials and furthermore<br />
Alu<br />
PMMA<br />
f [kHz]<br />
80<br />
60<br />
40<br />
20<br />
1 2 3 4 [cm] 5 6 7 8<br />
0<br />
0 0.5 1 1.5<br />
k<br />
B<br />
2 2.5 3<br />
Fig. 2. Final design corresponding to maximum relative band gap between<br />
third and fourth band (thick lines) for bending waves. Full<br />
lines: Longitudinal waves. Dashed lines: Bending waves.<br />
Table 3. Extreme frequency band values for bending waves in Fig. 2.<br />
Unit is kHz.<br />
k B ¼ 0 k B ¼ p<br />
f1 0 1.1508<br />
f2 5.2781 1.5692<br />
f3 5.9316 10.363<br />
f4 21.302 15.194<br />
f5 21.415 27.832<br />
f6 39.457 34.075<br />
has a pure 0/1 design. The majority of the cell is again<br />
occupied by the heavy and stiff material while a smaller<br />
part (around a fraction of 0.33) is occupied by the light<br />
and soft material. However, this time the light material is<br />
distributed in two different parts. The extreme frequency<br />
band values are listed in Table 3. The first three relative<br />
band gaps are listed in Table 6 (third column).<br />
As the last optimization example, the design resulting<br />
from maximizing the overlap between the first and second<br />
longitudinal band and the third and fourth bending band is<br />
shown in Fig. 3 together with the frequency bands. These<br />
band gaps are chosen because they have overlaps initially.<br />
An optimized base cell length of d ¼ 8:70 cm is found by<br />
the algorithm is this case (which is the reason for choosing<br />
this value in the previous two cases). The unique solution<br />
is a result of the frequency bands for the two wave<br />
types moving in the same direction but with different<br />
speeds as a function of base cell length. The design is<br />
quite similar to the design in the bending case. A larger<br />
part of the light material occupies the same amount of the<br />
cell as in the bending case, while the narrow part has decreased<br />
compared to the bending case. The extreme values<br />
for the different frequency bands are tabulated in Table 4.<br />
The relative band gaps for the separate wave types as well<br />
as for the combined problem are shown in Table 6 (fourth<br />
column) with F denoting the overlapping band gap for<br />
the combined problem. As expected, the first longitudinal