WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

3.4 Maximizing wave dissipation 23
a) b) c)
Figure 3.5 Optimized designs for the case of multi-directional wave input with a design
domain length of twice the wavelength of a shear wave. The design domain height equals
the length. Designs obtained for a) Pressure wave, b) Shear wave, c) Both pressure and
shear wave. From paper [5].
can excite high vibration levels.
Fig. 3.6 shows two examples of optimized distribution of two materials in a 1D
elastic bar and the corresponding vibration response. The response is recorded
in one end of the bar when the structure is subjected to a periodic load in the
other end 3 . The structures are obtained by formulating the optimization problem
so that the separation of two adjacent eigenfrequencies is maximized. The resulting
structures are clearly periodic-like bandgap structures with alternating sections of
the two materials corresponding to the design variables taking the values 0 and 1.
It is also evident that there exists a direct relation between the modes that are
separated and the number of material sections in the optimized structure.
Additionally, it turns out that there is a direct connection between a single
material contrast parameter and the maximum eigenfrequency separation that can
be obtained. Here, the separation is quantified as the ratio between two adjacent
eigenfrequencies. Fig. 3.7 shows the eigenfrequency separation for the optimized
structures versus the mode numbers that are separated. Depending only on the
contrast parameter β = ρ2E2/(ρ1E1), an asymptotic limit is found for how much
the eigenfrequencies can be separated. Here, E and ρ denote Young’s modulus and
mass density, respectively.
Itcanalsobeshownthatthefractionoccupiedbyeachofthetwomaterialsinthe
optimized structure relates directly to the contrast parameter α = ρ1E2/(ρ2E1). For
instance, it is observed that if α = 1 then the two materials are equally represented
in the optimized design.
3.4 Maximizing wave dissipation
All results presented so far have focused on the ability of bandgap (or periodic-like)
structures to quench vibrations or to reflect waves. However, another interesting
3 The 1D problem is supplemented by a similar problem in 2D in paper [7]. This is the contribution
of the co-author Niels L. Pedersen.