Sound transmission through a plate

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Silicon Nitride Membrane

Background
Benchmark ID	siliconNitrideMembrane_n60020m1q2
Category	misc
System-Class	AP-LTI-FOS
Parameters
nstates	60020
ninputs	1
noutputs	2
nparameters	2
components	A, B, C, E
Copyright
License	NA
Creator	Lihong Feng
Editor	Lihong Feng Christian Himpe
Location
NA

Figure 1: Sketch of the geometry.

Figure 2: Transfer function.

Description

The Sound transmission through a plate benchmark models the radiation of a vibrating plate and the excitation of a structure by an oscillating acoustic fluid. It is based on an experiment by Guy.

The system consists of a cuboid acoustic cavity, where one wall is considered a system of two parallel elastic brass plates with a $2\,\mathrm{cm}$ air gap between them; all other walls are considered rigid. The plates measure $0.2 \times 0.2\,\mathrm{m}$ and have a thickness of $t = 0.9144\,\mathrm{mm}$; the receiving cavity is $0.2\,\mathrm{m}$ wide. The outer plate is excited by a uniform pressure load and the resulting acoustic pressure in the receiving cavity is measured at the middle of the rigid wall opposite to the elastic plate ($P_1$ in the sketch).

The following material parameters have been considered for the brass plates and the acoustic fluid:

Part	Parameter	Value	Unit
Brass plates	$E$	$104$	$\mathrm{GPa}$
	$\rho$	$8500$	$\mathrm{kg}\,\mathrm{m}^{-3}$
	$\nu$	$0.37$	$-$
Acoustic fluid	$c$	$343$	$\mathrm{m}\,\mathrm{s}^{-1}$
	$\rho$	$1.21$	$\mathrm{kg}\,\mathrm{m}^{-3}$

Dimensions

System structure:

$\begin{align} M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\ y(t) &= C x(t) \end{align}$

System dimensions:

$M \in \mathbb{R}^{n \times n}$, $E \in \mathbb{R}^{n \times n}$, $K \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times 1}$, $C \in \mathbb{R}^{1 \times n}$, with $n=95\,480$.

Proportional damping, i.e. $E=\alpha M + \beta K$, with $\alpha=0, \beta=1\cdot 10^{-7}$ is considered. The two-way coupling between the structure and the acoustic fluid results in non-symmetric matrices $M, E, K$.

Data

The data is available at Zenodo.

Remarks

• The numerical model resembles the experimental data in a frequency range from $1\,\mathrm{Hz}$ to $1000\,\mathrm{Hz}$. The frequency response in this range is also included in the dataset.
• The finite element discretization has been performed with Kratos Multiphysics.
• The system has unstable eigenvalues. This is common in interior acoustic problems where no damping is assumed for the acoustic fluid.
• A comparison of different interpolation-based MOR methods using this benchmark example is available in

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

 @Misc{dataAum22,
   author =       {Aumann, Q.},
   title =        {Matrices for a sound transmission problem},
   howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
   year =         2022,
   doi =          {10.5281/zenodo.7300346}
 }

• For the background on the benchmark:

 @Article{AumW23,
   author =       {Aumann, Q. and Werner, S.~W.~R.},
   title =        {Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods},
   journal =      {Journal of Sound and Vibration},
   volume =       543,
   year =         2023,
   pages =        {117363},
   doi =          {10.1016/j.jsv.2022.117363},
   publisher =    {Elsevier {BV}}
 }