WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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Design variable, t
Design variable, t
1.0
0.5
0.0
0.00 0.25 0.50
Axial position
0.75 1.00
1.0
0.5
0.0
0.00 0.25 0.50
Axial position
0.75 1.00
ARTICLE IN PRESS
J.S. Jensen, N.L. Pedersen / Journal of Sound and Vibration 289 (2006) 967–986
40
20
0
-20
-40
-60
-80
0.0 2.0 4.0 6.0 8.0 10.0
Normalized frequency
comparison is shown in Fig. 3 with the results for the new objective function plotted with solid
lines and for the old objective function with dotted lines.
The material distribution curves left show that the new objective function has caused a shift in
the distribution between materials 1 and 2 and that the material in the end of the rods is now
material 2 instead of material 1. Interestingly, the response curves on the right show that the
response in the gaps optimized for maximum ratio drops lower in both examples even though the
absolute gap size is smaller for these designs.
We now introduce a material parameter b that characterizes the contrast between materials
b ¼ m Am B41, (15)
where the last inequality condition just implies that if not fulfilled the enumeration of the two
materials should be interchanged.
In Fig. 4, we show results for maximizing the ratio of eigenfrequencies for three different
combinations of material coefficients but keeping b ¼ 4. We vary m A and m B such that in the top
figures m A ¼ m B, in the middle figures m A ¼ 4, m B ¼ 1, and in the bottom figures m A ¼ 1 and
m B ¼ 4. For all three combinations we maximize the ratio for n ¼ 4 and find that the ratio for the
optimized designs in all cases becomes 3.09. For other combinations of material coefficients and
when optimizing for other n, we see that the maximum ratio between the adjacent
eigenfrequencies obtainable for any mode order seems to depend only on the parameter b.
However, as also seen in the figure, the material distribution varies and there is also a large
difference in the response curves for the different combinations of material coefficients.
Vel. response (dB)
Vel. response (dB)
50
0
-50
-100
-150
-200
-250
-300
0.0 10.0 20.0 30.0 40.0
Normalized frequency
Fig. 3. Comparison of optimized design (left) and velocity response (right) for maximizing the eigenfrequency gap
onþ1 on (dotted lines) and maximizing the eigenfrequency ratio o2 nþ1 =o2n (solid lines). Material parameters are as for
Fig. 2 and results are given for n ¼ 4 and 19.