WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
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Topological material layout in plates for vibration suppression and wave propagation control<br />
Fig. 8 Topology and energy transport for the optimization problem<br />
sketched in Fig. 7. Only energy vectors larger than 10% of<br />
the maximum vector are shown<br />
wave device. The simply supported plate is excited by<br />
two harmonic forces, where one is phase shifted by a<br />
quarter of a period. This will initiate a wave in the<br />
plate. The objective function is defined as the energy<br />
through three groups of elements as sketched in Fig. 7.<br />
This example is inspired by the working principles of<br />
piezoelectric ring motors that rely on ring waves and<br />
friction between a stator and a rotor, see e.g. Uchino<br />
(1998).<br />
In Fig. 8 the optimized structure is shown along<br />
with the energy transport. We see that energy is being<br />
guided in a near-circular path as defined in the optimization<br />
problem but also that some energy escapes<br />
this path. This can for instance be seen in the lower right<br />
part of the plate. Especially at the lower optimization<br />
area we see that the energy vectors have a relatively<br />
large vertical component while in the other two areas<br />
the energy is more efficiently guided only in the desired<br />
direction.<br />
8 Conclusion<br />
We have presented a material layout algorithm based<br />
on the method of topology optimization. The algorithm<br />
allows for the design of bi-material elastic plates with<br />
optimized vibrational suppression and guided energy<br />
transport when subjected to harmonic excitation.<br />
The optimization algorithm is based on a finite element<br />
model of plate vibration for moderately thick<br />
plates using the Mindlin plate theory. Special consideration<br />
is given to the derivation of an energy transport<br />
measure using the Poynting vector.<br />
The optimization algorithm is based on the FE<br />
model combined with analytical sensitivity analysis performed<br />
with the adjoint method. An iterative optimization<br />
scheme is set up with design updates performed<br />
with the method of moving asymptotes.<br />
To demonstrate the capabilities of the method we<br />
have presented two sets of examples. The first deals<br />
with minimizing the total vibrational response of the<br />
plate in response to harmonic excitation at one or<br />
several distinct frequencies. The resulting bi-material<br />
designs display a low vibrational response at the target<br />
frequency range by moving the plate natural frequencies<br />
away from these targets.<br />
The second, and more challenging, set of examples<br />
deals with optimizing the materials distribution in order<br />
to direct the vibrational energy flow in specified paths<br />
in the plate. It was demonstrated that the specified<br />
energy transporting behavior could be realized by a<br />
complicated material layout and the rather difficult task<br />
of creating a ring wave device that functions by material<br />
contrasts was accomplished.<br />
Acknowledgements This work received support from the<br />
Eurohorcs/ESF European Young Investigator Award (www.esf.<br />
org/euryi) through the grant “Synthesis and topology optimization<br />
of optomechanical systems”, the New Energy and Industrial<br />
Technology Development Organization project (Japan), and<br />
from the Danish Center for Scientific Computing.<br />
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