Broadband vibration attenuation from a one-dimensional acoustic black hole resonator for plate-on-plate structures

Abstract

This paper investigates the vibrational behaviour and attenuation performance of a plate connected to a plate-like resonator equipped with a 1D acoustic black hole. The ABH principle relies on trapping the incident waves and consequently absorbing the incoming mechanical energy. Conversely, the ABH acts as a conventional dynamic absorber in which the vibration modes of the wedge act as a multifrequency resonator when attached to a host structure. The typically added damping layer enriches its dynamic, providing superior attenuation across a broader frequency band. This plate-like resonator can be used as a vibration control alternative to the use of single or multiple resonators attached to the host structure. In this work, a mobility-based approach is used to connect both the host plate and the plate-like resonator with an ABH termination. The response of both structures is given by a Finite Element model implemented by the authors in MATLAB, and the point connections are assumed to be rigid. The results are compared to that of a uniform plate acting as a multifrequency resonator, and it is shown that the ABH-based vibration attenuation leads to a reduction of up to 20 dB across multiple third-octave bands in the mobility response of the coupled system. Additionally, the energy flow is also investigated, and it is shown that the ABH-plate resonator yields up to 50 dB reduction also over multiple third-octave bands in the spatially averaged kinetic energy spectra. The addition of the film of viscoelastic material enhances its attenuation performance by up to 30 dB compared to the geometric effect due to the ABHs. The results suggest that this ABH-based solution can be a viable engineering approach and opens the way for new and innovative solutions in vibration control.

Appendix: Mass and Stiffness matrices for the ACM element

A piecewise constant approach was considered for the thickness of each element of the ABH resonator. Please note that where the thickness $h$ is mentioned, this value is given thickness at the central point of the element, following the decay along the ABH termination.

$$ \left[ {\mathbf{m}} \right]_{e} = \frac{\rho hab}{{6300}}\left[ {\begin{array}{*{20}c} {{\mathbf{m}}_{11} } & {{\mathbf{m}}_{21}^{T} } \\ {{\mathbf{m}}_{21} } & {{\mathbf{m}}_{22} } \\ \end{array} } \right] $$

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where $\left[ {\mathbf{m}} \right]_{e}$ is the element inertia matrix, $\rho$ is the density, $h$ is the plate’s thickness, and $2a$ and $2b$ are the size of the element’s edges.

$$ {\mathbf{m}}_{{11}} = \left[ {\begin{array}{*{20}l} {3454} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {922b} \hfill & {320b^{2} } \hfill & {} \hfill & {} \hfill & {} \hfill & {{\text{Sym}}} \hfill \\ { - 922a} \hfill & { - 252ab} \hfill & {320a^{2} } \hfill & {} \hfill & {} \hfill & {} \hfill \\ {1226} \hfill & {398b} \hfill & { - 548a} \hfill & {3454} \hfill & {} \hfill & {} \hfill \\ {398b} \hfill & {160b^{2} } \hfill & {160b^{2} } \hfill & {922b} \hfill & {320b^{2} } \hfill & {} \hfill \\ {548a} \hfill & {168ab} \hfill & {240a^{2} } \hfill & {922a} \hfill & {252ab} \hfill & {320a^{2} } \hfill \\ \end{array} } \right] $$

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$$ {\mathbf{m}}_{{22}} = \left[ {\begin{array}{*{20}l} {3454} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ { - 922b} \hfill & {320b^{2} } \hfill & {} \hfill & {} \hfill & {} \hfill & {{\text{Sym}}} \hfill \\ {922{\text{a}}} \hfill & { - 252ab} \hfill & {320a^{2} } \hfill & {} \hfill & {} \hfill & {} \hfill \\ {1226} \hfill & { - 398b} \hfill & {548a} \hfill & {3454} \hfill & {} \hfill & {} \hfill \\ { - 398b} \hfill & {160b^{2} } \hfill & { - 168ab} \hfill & { - 922b} \hfill & {320b^{2} } \hfill & {} \hfill \\ { - 548a} \hfill & {168ab} \hfill & { - 240a^{2} } \hfill & { - 922a} \hfill & {252ab} \hfill & {320a^{2} } \hfill \\ \end{array} } \right] $$

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$$ {\mathbf{m}}_{{21}} = \left[ {\begin{array}{*{20}l} {394} \hfill & {232b} \hfill & { - 232a} \hfill & {1226} \hfill & {548b} \hfill & {398a} \hfill \\ { - 232b} \hfill & { - 120b^{2} } \hfill & {112ab} \hfill & { - 548b} \hfill & { - 240b^{2} } \hfill & { - 168ab} \hfill \\ {232a} \hfill & {112ab} \hfill & { - 12a^{2} } \hfill & {398a} \hfill & {168ab} \hfill & {160a^{2} } \hfill \\ {1226} \hfill & {548b} \hfill & { - 398a} \hfill & {394} \hfill & {232b} \hfill & {232a} \hfill \\ { - 548b} \hfill & { - 240b^{2} } \hfill & {168ab} \hfill & { - 232b} \hfill & { - 120b^{2} } \hfill & { - 112ab} \hfill \\ { - 398a} \hfill & { - 168ab} \hfill & {160a^{2} } \hfill & { - 232a} \hfill & { - 112ab} \hfill & { - 120a^{2} } \hfill \\ \end{array} } \right] $$

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$$ \left[ {\mathbf{k}} \right]_{e} = \frac{{Eh}}{{48\left( {1 - \nu ^{2} } \right)ab}}\left[ {\begin{array}{*{20}l} {{\mathbf{k}}_{{11}} } \hfill & {} \hfill & {{\text{sym}}} \hfill & {} \hfill \\ {{\mathbf{k}}_{{21}} } \hfill & {{\mathbf{k}}_{{22}} } \hfill & {} \hfill & {} \hfill \\ {{\mathbf{k}}_{{31}} } \hfill & {{\mathbf{k}}_{{32}} } \hfill & {{\mathbf{k}}_{{33}} } \hfill & {} \hfill \\ {{\mathbf{k}}_{{41}} } \hfill & {{\mathbf{k}}_{{42}} } \hfill & {{\mathbf{k}}_{{43}} } \hfill & {{\mathbf{k}}_{{44}} } \hfill \\ \end{array} } \right] $$

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where $\left[ {\mathbf{k}} \right]_{e}$ is the element stiffness matrix, $E$ is plate’s Young’s Modulus, $h$ is the thickness of the plate, $\nu$ is the Poisson’s ratio, and $2a$ and $2b$ are the size of the element’s edges.

$$ {\mathbf{k}}_{11} = \left[ {\begin{array}{*{20}c} {4\left( {\beta^{2} + \alpha^{2} } \right) + \frac{2}{5}\left( {7 - 2\nu } \right) } & {} & {{\text{Sym}}} \\ {2\left[ {2\alpha^{2} + \frac{1}{5}\left( {1 + 4\nu } \right)} \right]b} & {4\left[ {\frac{4}{3}\alpha^{2} + \frac{4}{15}\left( {1 - \nu } \right)} \right]b^{2} } & {} \\ {2\left[ { - 2\beta^{2} - \frac{1}{5}\left( {1 + 4\nu } \right)} \right]a} & { - 4\nu ab} & {4\left[ {\frac{4}{3}\beta^{2} + \frac{4}{15}\left( {1 - \nu } \right)} \right]a^{2} } \\ \end{array} } \right] $$

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$$ {\mathbf{k}}_{21} = \left[ {\begin{array}{*{20}c} { - \left[ {2\left( {2\beta^{2} - \alpha^{2} } \right) + \frac{2}{5}\left( {7 - 2\nu } \right)} \right] } & {2\left[ {\alpha^{2} - \frac{1}{5}\left( {1 + 4\nu } \right)} \right]b} & {2\left[ {2\beta^{2} + \frac{1}{5}\left( {1 - \nu } \right)} \right]a} \\ {2\left[ {\alpha^{2} - \frac{1}{5}\left( {1 + 4\nu } \right)} \right]b} & {4\left[ {\frac{2}{3}\alpha^{2} - \frac{4}{15}\left( {1 - \nu } \right)} \right]b^{2} } & 0 \\ { - 2\left[ {2\beta^{2} + \frac{1}{5}\left( {1 - \nu } \right)} \right]a} & 0 & {4\left[ {\frac{2}{3}\beta^{2} - \frac{1}{15}\left( {1 - \nu } \right)} \right]a^{2} } \\ \end{array} } \right]{ } $$

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$$ {\mathbf{k}}_{31} = \left[ {\begin{array}{*{20}c} { - \left[ {2\left( {2\beta^{2} + \alpha^{2} } \right) - \frac{2}{5}\left( {7 - 2\nu } \right)} \right] } & {2\left[ { - \alpha^{2} + \frac{1}{5}\left( {1 - \nu } \right)} \right]b} & {2\left[ {\beta^{2} - \frac{1}{5}\left( {1 - \nu } \right)} \right]a} \\ {2\left[ {\alpha^{2} - \frac{1}{5}\left( {1 - \nu } \right)} \right]b} & {4\left[ {\frac{1}{3}\alpha^{2} + \frac{1}{15}\left( {1 - \nu } \right)} \right]b^{2} } & 0 \\ {2\left[ { - \beta^{2} + \frac{1}{5}\left( {1 - \nu } \right)} \right]a} & 0 & {4\left[ {\frac{1}{3}\beta^{2} + \frac{1}{15}\left( {1 - \nu } \right)} \right]a^{2} } \\ \end{array} } \right] $$

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$$ {\mathbf{k}}_{41} = \left[ {\begin{array}{*{20}c} {2\left( {\beta^{2} - 2\alpha^{2} } \right) - \frac{2}{5}\left( {7 - 2\nu } \right) } & {2\left[ { - 2\alpha^{2} - \frac{1}{5}\left( {1 - \nu } \right)} \right]b} & {2\left[ { - \beta^{2} + \frac{1}{5}\left( {1 + 4\nu } \right)} \right]a} \\ {2\left[ {2\alpha^{2} + \frac{1}{5}\left( {1 - \nu } \right)} \right]b} & {4\left[ {\frac{2}{3}\alpha^{2} - \frac{1}{15}\left( {1 - \nu } \right)} \right]b^{2} } & 0 \\ {2\left[ { - \beta^{2} + \frac{1}{5}\left( {1 + 4\nu } \right)} \right]a} & 0 & {4\left[ {\frac{2}{3}\beta^{2} - \frac{4}{15}\left( {1 - \nu } \right)} \right]a^{2} } \\ \end{array} } \right] $$

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where $\alpha = a/b$ and $\beta = b/a$.

$$ \begin{array}{*{20}c} {{\mathbf{k}}_{22} = {\mathbf{I}}_{3}^{T} {\mathbf{k}}_{11} {\mathbf{I}}_{3} ;} & {} & {} \\ {{\mathbf{k}}_{32} = {\mathbf{I}}_{3}^{T} {\mathbf{k}}_{41} {\mathbf{I}}_{3} ;} & {{\mathbf{k}}_{33} = {\mathbf{I}}_{1}^{T} {\mathbf{k}}_{11} {\mathbf{I}}_{1} ;} & {} \\ {{\mathbf{k}}_{42} = {\mathbf{I}}_{3}^{T} {\mathbf{k}}_{31} {\mathbf{I}}_{3} ;} & {{\mathbf{k}}_{43} = {\mathbf{I}}_{1}^{T} {\mathbf{k}}_{21} {\mathbf{I}}_{1} ;} & {{\mathbf{k}}_{44} = {\mathbf{I}}_{2}^{T} {\mathbf{k}}_{11} {\mathbf{I}}_{2} ;} \\ \end{array} $$

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where ${\mathbf{I}}_{1} = \left[ {\begin{array}{*{20}c} { - 1} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]$, ${\mathbf{I}}_{2} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & { - 1} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]$ and ${\mathbf{I}}_{3} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & { - 1} \\ \end{array} } \right]$.