Metamaterials

The term "metamaterial" describes a material that has properties which cannot be observed in nature. The main concept of those materials is the arrangement of repeating sub elements made of conventional material. The periodic alignment of those sub elements made of conventional material can result in completely different macroscopical material behavior. In many examples, metamaterials have dynamically modified properties as negative density, negative compressibility, or negative refraction. This special behavior offers a wide range of revolutionary applications in vibroacoustics.

Inserting local resonances into a structural component, it is possible to influence the wave propagation for specific frequencies. Designing the unitcell and the host structure, it is possible to tailor the vibroacoustic properties of the structure.

In this context, the term "stop band" is important and therefore often used. It describes the frequency range where no free wave propagation is observed. This is induced by destructive interference between transmitted and reflected waves or local resonances.

Numerical Invesitgations

Homogeneous Beam

Evaluation of dispersion characteristics of homogeneous beam of infinite length using the WFEM:

Wave solution:

$\begin{align*} \hat{w}(x,t) &= \hat{w}_0 \; e^{-i(k(x-ct))}\\ &= \hat{w}_0 \; e^{-ikx} \; e^{ikct} \\ &= \hat{w}_0 \; e^{-ikx} \; e^{i\Omega t} \end{align*}$

with wavenumber $\begin{array}{l}k\end{array}$ and circular frequency $\begin{array}{l}\Omega = kc\end{array}$ . Thus $\begin{array}{l}c = \frac{\Omega}{k}\end{array}$ .

For a homogeneous beam no stop band is observed.

$\begin{align*} \text{Frequency:} \; \; \; &f = \frac{\Omega}{2\pi} \\ \text{Phase:} \; \; \; &\varphi = k \; L_x \\ & c = \frac{\Omega}{k} \end{align*}$

$\begin{array}{l}\displaystyle \begin{equation*} c_L = \sqrt{E/\rho} = 5093 \left[\frac{\text{m}}{\text{s}}\right] \; \approx \; \frac{2 \pi \; \Delta f \; L_x}{\Delta \varphi} = \frac{2 \pi * 2*10^4 * 0.06}{1.5} = 5027 \left[\frac{\text{m}}{\text{s}}\right] \end{equation*}\end{array}$

Inclusions

Evaluation of influence of inclusions on dispersion characteristics:

For inclusions of equal size the stop band width is influenced by the inclusion shape.

Tuned Mass Damper (TMD)

Evaluation of influence of TMDs aligned on homogeneous beam of infinite length:

$\begin{array}{l}\displaystyle \begin{equation*} f_E = 8457 \; \; [\text{Hz}] \end{equation*}\end{array}$

Evaluation of influence of TMD alignment for a TMD which is installed into an inclusion:

$\begin{array}{l}\displaystyle \begin{equation*} f_E = 8457 \; \; [\text{Hz}] \end{equation*}\end{array}$

Advanced Applications

Porous Metamaterials

Porous materials can have favorable absorption characteristics However, the thickness of the porous layer with respect to the signal wavelength is decisive for the absorption Hence, an idea is to reduce the necessary thickness by embedding structures in the porous layer.

Acoustic Black Holes

Using local thickness variations, it is possible to modulate bending waves that propagate in thin walled structures.

$\begin{align*} h(x) &\rightarrow 0 \\ A(x) &\rightarrow \infty \\ c(x) &\rightarrow 0 \end{align*}$

Conclusion

Dispersion characteristics of structures can be manipulated by inclusions and TMDs
By shaping a structure appropriately stop bands can be induced
If a TMD is installed into an inclusion the wave propagation in the structure is influenced by the alignment of the TMD

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