Reference problem
=====================




Geometry
---------

Rectangular beam, composed of two layers of different materials:


.. image:: images/Cadre1.gif


.. _RefSchema_Cadre1.gif:

Image 1.1-1: Geometry of the viscoelastic beam.

Width: 0.01 m

Length: 0.15 m

Thickness: Elastic material (layer No. 1 at the bottom): 0.001 m

 Viscoelastic material (top layer no. 2): 0.002 m




Material properties
------------------------

The material of layer No. 1 below is isotropic elastic (steel); its properties are constant:


* Young's modulus :math:`E=210000\mathit{MPa}`


* Poisson's ratio :math:`\nu =\mathrm{0,3}`


* density :math:`\rho =7800\mathit{kg}/{m}^{3}`


* hysteretic damping :math:`\eta =\mathrm{0,001}`

The material of the top layer No. 2 is viscoelastic (elastomer); some of its properties are frequency-dependent:


.. csv-table::

    "Frequency (Hz)", "Real part of Young's modulus :math:`E` (MPa)", "Loss factor :math:`\eta`"
    "1", "23.2", "1.1"
    "10", "58", "0.85"
    "50", "145", "0.7"
    "100", "203", "0.6"
    "500", "348", "0.4"
    "1000", "435", "0.35"
    "1500", "464", "0.34"


**Table** 1.2-1 **:** ****Frequency-dependent properties of viscoelastic material. **

The others are constant:


* Poisson's ratio :math:`\nu =\mathrm{0,45}`


* density :math:`\rho =1200\mathit{kg}/{m}^{3}`




Boundary conditions and loads
-------------------------------------

Embedded on a steel edge.




Initial conditions
--------------------

Not applicable (natural mode calculation).