Plate with tuned vibration absorbers

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Figure 1: Sketch of the geometry.

Figure 2: Frequency response function.

Description

The Plate with tuned vibration absorbers benchmark models the vibration response of a plate excited by a single harmonic load. The plate is equipped with 108 tuned vibration absorbers (TVA), which change the vibration pattern of the host structure in a narrow frequency band around their tuning frequency. Such systems have, for example, been examined in and .

This benchmark models an aluminum plate with dimensions $0.8 \times 0.8\,\mathrm{m}$ and a thickness of $t = 1\,\mathrm{mm}$. The surrounding edges are simply supported. The plate's surface is equipped with six struts along the $y$-direction, on which the TVAs are placed. The TVAs have a combined mass of $10\,\%$ of the plate mass and are tuned to $f=48\,\mathrm{Hz}$. The plate is excited by a single load near one of the corners of the plate (see sketch). The root mean square of the displacement in $z$-direction at all points of the plate surface is plotted in Figure 2. The effect of the TVAs is clearly visible in the frequency range around their tuning frequency.

The following material parameters have been considered for aluminum:

Parameter	Value	Unit
$E$	$69$	$\mathrm{GPa}$
$\rho$	$2650$	$\mathrm{kg}\,\mathrm{m}^{-3}$
$\nu$	$0.22$	$-$

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:

$M \in \mathbb{R}^{n \times n}$, $E \in \mathbb{R}^{n \times n}$, $K \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times 1}$, $C \in \mathbb{R}^{q \times n}$, with $n=201\,900$ and $q=28\,278$.

Proportional damping, i.e. $E=\alpha M + \beta K$, with $\alpha=1\cdot 10^{-2}, \beta=1\cdot 10^{-4}$ is considered. The matrices $M, E, K$ are positive (semi-) definite.

Data

The data is available at Zenodo.

Remarks

• The dataset also contains a version with a single output (SISO). Here, the displacement of the plate at the location of the load is evaluated.
• The frequency response in the range $1\,\mathrm{Hz}$ to $250\,\mathrm{Hz}$ is included in the dataset.
• The finite element discretization has been performed with Kratos Multiphysics.
• 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{dataAum23,
   author =       {Aumann, Q.},
   title =        {Matrices for a plate with tuned vibration absorbers},
   howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
   year =         2023,
   doi =          {10.5281/zenodo.7507011}
 }

• 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}}
 }