WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
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Inverse design of phononic crystals by tobology optimization 899<br />
PMMA<br />
f [kHz]<br />
band gap is smaller than the corresponding first (optimized)<br />
relative band gap for the single longitudinal case, while the<br />
second and third are actually larger. Also, the third relative<br />
band gap for bending waves is smaller (but not much) than<br />
the corresponding third (optimized) relative band gap for the<br />
single bending case. The similarity between this design and<br />
the bending case design (they are almost eachother’s mirrors)<br />
is also reflected in their performances, which are very close<br />
(relative band gaps of 0.374 and 0.378).<br />
Torsion waves in a beam are described by the same<br />
differential equation as for longitudinal waves, but with<br />
E K<br />
the factor<br />
replacing E. Here I is the area mo-<br />
2ð1 þ nÞ I<br />
ment of inertia as before and K is the torsion stiffness<br />
cross-sectional factor. For a circular cross-section K ¼ I<br />
and it may be seen that the torsional frequencies are lower<br />
than<br />
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi<br />
the longitudinal frequencies by a factor of<br />
2ð1 þ nÞ 1:6. Comparing the torsional frequency<br />
bands with the bands in Fig. 3 for the optimized design in<br />
the combined case shows that there is a band gap between<br />
10.6–13.2 kHz (relative band gap 0.43) for all three wave<br />
types, i.e. in this interval all three wave types are prevented<br />
from propagating in the beam.<br />
Since for all geometries K<br />
1 (with “¼” only for cir-<br />
I<br />
cular cross-sections), it is not possible to obtain<br />
K<br />
I<br />
Alu<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. 3. Final design corresponding to maximum relative band gap between<br />
first and second band for longitudinal waves and third and<br />
fourth band for bending waves. Full lines: Longitudinal waves.<br />
Dashed lines: Bending waves.<br />
Table 4. Extreme frequency band values for bending and longitudinal<br />
waves in Fig. 3. Unit is kHz.<br />
B waves L waves<br />
k B ¼ 0 k B ¼ p k B ¼ 0 k B ¼ p<br />
f1 0 1.145 0 10.643<br />
f2 5.6165 1.6585 27.272 21.993<br />
f3 5.9774 10.643 42.172 45.188<br />
f4 21.454 15.534 77.451 72.706<br />
¼ 2ð1 þ nÞ which would result in perfectly coinciding<br />
[dB/1.00 (m/s†)/N]<br />
dB<br />
80<br />
60<br />
40<br />
20<br />
0<br />
-20<br />
FRF (Magnitude)<br />
Working : PMMA-Alu-11-seg-7.5cm-200302-ref : Input : FFT Analyzer<br />
0 2k 4k 6k 8k 10k 12k 14k<br />
[Hz]<br />
16k 18k 20k 22k 24k 26k<br />
80<br />
70<br />
60<br />
50<br />
40<br />
30<br />
20<br />
10<br />
0<br />
-10<br />
-20<br />
-30<br />
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000<br />
Hz<br />
Fig. 4. Frequency response curves from [13]. Top: Experimental. Bottom:<br />
Theoretical.<br />
Table 5. Frequency band gaps in Fig. 4 determined by inspection.<br />
Unit is kHz.<br />
f min<br />
1<br />
f max<br />
1<br />
f min<br />
2<br />
f max<br />
2<br />
5.1 13.1 15.4 24.3<br />
torsional and longitudinal bands. To increase the common<br />
gap size, one may consider lowering the mid-gap values<br />
between higher torsional modes. This can be obtained by<br />
decreasing the ratio K<br />
which can be achieved by choosing<br />
I<br />
a non-circular cross-section. This aspect will be investigated<br />
in future work.<br />
In the following, the above results are compared with<br />
the experimental results presented in [13]. Here, the set-up<br />
is a (non-optimized) bar made up of five-and-a-half repetitions<br />
of a base section consisting of two bars glued together<br />
with circular cross-sections with diameter 1 cm and<br />
made of PMMA and Aluminum respectively, as in the<br />
above case. Each bar is 7.5 cm resulting in a base cell<br />
with a length of 15 cm. An experimental response curve<br />
showing the acceleration response at the end of the bar as<br />
a function of the longitudinal vibration excitation frequency<br />
applied to the opposite end is shown in Fig. 4<br />
(upper) with a corresponding theoretical prediction shown<br />
below. The 40 dB horizontal line has been used to identify