WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Inverse design of phononic crystals by tobology optimization 899
PMMA
f [kHz]
band gap is smaller than the corresponding first (optimized)
relative band gap for the single longitudinal case, while the
second and third are actually larger. Also, the third relative
band gap for bending waves is smaller (but not much) than
the corresponding third (optimized) relative band gap for the
single bending case. The similarity between this design and
the bending case design (they are almost eachother’s mirrors)
is also reflected in their performances, which are very close
(relative band gaps of 0.374 and 0.378).
Torsion waves in a beam are described by the same
differential equation as for longitudinal waves, but with
E K
the factor
replacing E. Here I is the area mo-
2ð1 þ nÞ I
ment of inertia as before and K is the torsion stiffness
cross-sectional factor. For a circular cross-section K ¼ I
and it may be seen that the torsional frequencies are lower
than
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
the longitudinal frequencies by a factor of
2ð1 þ nÞ 1:6. Comparing the torsional frequency
bands with the bands in Fig. 3 for the optimized design in
the combined case shows that there is a band gap between
10.6–13.2 kHz (relative band gap 0.43) for all three wave
types, i.e. in this interval all three wave types are prevented
from propagating in the beam.
Since for all geometries K
1 (with “¼” only for cir-
I
cular cross-sections), it is not possible to obtain
K
I
Alu
80
60
40
20
1 2 3 4 5 6 7 8
[cm]
0
0 0.5 1 1.5
k
B
2 2.5 3
Fig. 3. Final design corresponding to maximum relative band gap between
first and second band for longitudinal waves and third and
fourth band for bending waves. Full lines: Longitudinal waves.
Dashed lines: Bending waves.
Table 4. Extreme frequency band values for bending and longitudinal
waves in Fig. 3. Unit is kHz.
B waves L waves
k B ¼ 0 k B ¼ p k B ¼ 0 k B ¼ p
f1 0 1.145 0 10.643
f2 5.6165 1.6585 27.272 21.993
f3 5.9774 10.643 42.172 45.188
f4 21.454 15.534 77.451 72.706
¼ 2ð1 þ nÞ which would result in perfectly coinciding
[dB/1.00 (m/s†)/N]
dB
80
60
40
20
0
-20
FRF (Magnitude)
Working : PMMA-Alu-11-seg-7.5cm-200302-ref : Input : FFT Analyzer
0 2k 4k 6k 8k 10k 12k 14k
[Hz]
16k 18k 20k 22k 24k 26k
80
70
60
50
40
30
20
10
0
-10
-20
-30
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000
Hz
Fig. 4. Frequency response curves from [13]. Top: Experimental. Bottom:
Theoretical.
Table 5. Frequency band gaps in Fig. 4 determined by inspection.
Unit is kHz.
f min
1
f max
1
f min
2
f max
2
5.1 13.1 15.4 24.3
torsional and longitudinal bands. To increase the common
gap size, one may consider lowering the mid-gap values
between higher torsional modes. This can be obtained by
decreasing the ratio K
which can be achieved by choosing
I
a non-circular cross-section. This aspect will be investigated
in future work.
In the following, the above results are compared with
the experimental results presented in [13]. Here, the set-up
is a (non-optimized) bar made up of five-and-a-half repetitions
of a base section consisting of two bars glued together
with circular cross-sections with diameter 1 cm and
made of PMMA and Aluminum respectively, as in the
above case. Each bar is 7.5 cm resulting in a base cell
with a length of 15 cm. An experimental response curve
showing the acceleration response at the end of the bar as
a function of the longitudinal vibration excitation frequency
applied to the opposite end is shown in Fig. 4
(upper) with a corresponding theoretical prediction shown
below. The 40 dB horizontal line has been used to identify