WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

976
Eigenvalue ratio, ω n+1 2 /ωn 2
20
15
10
5
ARTICLE IN PRESS
J.S. Jensen, N.L. Pedersen / Journal of Sound and Vibration 289 (2006) 967–986
0
1 10
Mode number, n
100
Fig. 5. The maximum eigenfrequency ratio o2 nþ1 =o2n as a function of the mode order n. Curves are presented for the
homogeneous case (’) as well as for three values of b ¼ mAmB; b ¼ 2(n) b ¼ 9(&), and b ¼ 29:7 ( ), the latter
corresponding to the elastic rod case.
Volume fraction (material 2)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.01 0.1 1
Material parameter, α
10 100
Fig. 6. Fraction of material 2 in the optimized design as a function of the material parameter a ¼ m A=m B. The fraction is
calculated in the middle of the structure where the end effects do not play a role.
coefficients. In order to analyze this effect, a second material parameter is introduced:
a ¼ mA . (16)
mB We now optimize the ratio for different material coefficients but keep a constant. In this case the
optimized material distribution is always the same, whereas the maximum eigenvalue ratio varies
significantly. In Fig. 6, we plot the volume fraction of material 2 versus the value of a. The volume
fraction is computed in the interior part of the domain where the boundary effects are not
important. As seen, with a ¼ 1 the two materials are evenly distributed in the optimized design,