WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

3.1 Vibration-quenching structures 19
a) b) c)
Figure 3.1 Optimized distribution of two elastic materials in a 2D structure (12×12cm)
for creating minimum responseat 63kHz. a) Structural domain andloading, b) Optimized
distribution for a high material contrast, c) Optimized distribution for a low material
contrast. In four considered scenarios, the load acts on one edge and the response is
minimized on the opposite edge. From paper [4].
two materials with a high contrast between their material properties (mass density
and Young’s modulus). Fig. 3.1c shows the distribution of the materials if the
contrast is smaller. Four different loading scenarios are considered (with loading on
each boundary) so that symmetrical structures are obtained.
For the high contrast case the structure appears periodic-like. However, it must
be observed that the periodicity is irregular and changes near the free edges of the
structure. This is not surprising, considering the results obtained in Section 2.1
which demonstrate that waves propagate differently along the structural boundary
due to the presence of a free edge. Thus, along a free edge a different periodicity is
needed to quench the vibrations. If the contrast is smaller, the structure cannot be
characterized as periodic-like – although periodic substructures are observed. This
is due to the fact that for the low contrast no bandgaps can be created for the
infinite material 1 and consequently a periodic-like finite structure does not emerge.
Fig. 3.2 shows the vibration response curves for the two optimized structures.
It is noticed that the response for the high contrast case has been significantly
reduced. A clear bandgap footprint is observed with a well-defined low response
frequency range. The vibration reduction is not nearly as pronounced for the low
contrast case. It resembles more the low response obtained at the anti-resonances
that distributed throughout the displayed frequency range. Furthermore, the figures
show the response computed with a large amount of mass-proportional damping
present. The significance of the damping is explained in the following.
The first attempts to find optimized material distributions for the structure in
Fig. 3.1 were unsuccessful. Amessy material distribution was obtained with no clear
1 A contribution from co-author Ole Sigmund.