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


Geometry
---------


.. image:: images/100002000000033B0000010F68EF68BB5B4F08C5.png
    :width: 6.4236in
    :height: 2.0681in


.. _RefImage_100002000000033B0000010F68EF68BB5B4F08C5.png:





**Figure 1.1 Problem geometry and loading system**

Geometry of the :math:`(m)` beam:

 :math:`L=2.436`

 :math:`R=0.00795`

 :math:`r=0.00680`


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

Beam

.. csv-table::

    ":math:`E=2.07\mathrm{E11}\mathrm{Pa}` ", "Young's module"
    ":math:`\nu =0.3` ", "Poisson's ratio"
    ":math:`\rho =7870.0{\mathrm{kg.m}}^{-3}` ", "Density"
    ":math:`\mathrm{AMOR}\text{\_}\mathrm{ALPHA}\text{}=\text{}1.79E-5{\mathrm{N.s.m}}^{-1}` ", ""
    ":math:`\mathrm{AMOR}\text{\_}\mathrm{BETA}\text{}=\text{}0.1526{\mathrm{N.kg}}^{-1}` ", ""


The coefficients :math:`\alpha` and :math:`\beta` make it possible to build a viscous damping matrix proportional to stiffness and mass :math:`[C]=\alpha [K]+\beta [M]`

Obstacles

.. csv-table::

    ":math:`\mathit{RIGI}\text{\_}\mathit{NOR}=1.0E5{\mathit{N.m}}^{-1}` ", "normal stiffness coefficient"
    ":math:`\mathrm{AMOR}\text{\_}\mathrm{NOR}=0.28{\mathrm{N.m.s}}^{-1}` ", "normal damping coefficient"



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

Imposed displacement:

.. csv-table::

    "All nodes on the beam:", ":math:`\mathrm{DZ}=0`,", :math:`\mathrm{DRY}=0`, :math:`\mathrm{DRX}=0`"
    "Node :math:`\mathrm{N1}`:", ":math:`\mathrm{DX}=0`, :math:`\mathrm{DY}=0`, :math:`\mathrm{DRZ}=0`"


Imposed load :math:`(N)`:

.. csv-table::

    "Knots :math:`\mathrm{N3}` to :math:`\mathrm{N13}` and :math:`\mathrm{N27}` to :math:`\mathrm{N37}` "," :math:`\mathrm{FORCEP}=4.138\text{}\mathrm{sin}(\omega t)`"
    "Knots :math:`\mathrm{N5}` to :math:`\mathrm{N25}` and :math:`\mathrm{N39}` to :math:`\mathrm{N49}` "," :math:`\mathrm{FORCEM}=-4.138\text{}\mathrm{sin}(\omega t)`"


with :math:`\omega =251.2{\mathrm{rad.s}}^{-1}(40\mathrm{Hz})`

Obstacles located in plane :math:`Y` following direction :math:`y`:

.. csv-table::

    ":math:`\mathrm{N14}` ", "Game = :math:`0.406E-3m` ", "origin = :math:`(0.609\mathrm{,0}.0\mathrm{,0}.0)`"
    ":math:`\mathrm{N26}` ", "Game = :math:`0.406E-3m` ", "origin = :math:`(1.218\mathrm{,0}.0\mathrm{,0}.0)`"
    ":math:`\mathrm{N38}` ", "Game = :math:`0.406E-3m` ", "origin = :math:`(1.827\mathrm{,0}.0\mathrm{,0}.0)`"
    ":math:`\mathrm{N2}` ", "Game = :math:`0.406E-3m` ", "origin = :math:`(2.436\mathrm{,0}.0\mathrm{,0}.0)`"