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


Geometry
---------

Two problems are considered (1 and 2), corresponding respectively to models A and B. The aim is to calculate the natural modes of a supported and supported beam (problem 1) and free-free (problem 2), with a plane modeling.


.. image:: images/100002010000050600000171F61441F7BAE88C7D.png
    :width: 5.4882in
    :height: 1.5752in


.. _RefImage_100002010000050600000171F61441F7BAE88C7D.png:

For problem 2 the beam is identical to problem 1 but without the supports at points :math:`A` and :math:`C`.

Dimensions of the **rectangular straight beam:**


.. csv-table::

    "diameter:", ":math:`R=0.05m`"
    "length:", ":math:`L=6.m`"


**Point coordinates** (:math:`m`) **:**


.. csv-table::

    "", ":math:`A` "," :math:`B` "," :math:`C`"
    ":math:`x` ", "0. ", "3. ", "6."
    ":math:`y` and :math:`z` ", "0. ", "0. ", "0."



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

:math:`E={2.10}^{11}\mathrm{Pa}`

:math:`\nu =0.3`

:math:`\rho =7800.\mathrm{kg}/{m}^{3}`


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


.. csv-table::

    "Problem 1:", "Point :math:`A` "," :math:`\mathit{DX}=\mathit{DY}=0.` ", "Point :math:`C` "," :math:`\mathit{DY}=0.`"
    "Problem 2:", "Point :math:`A` ", "", "Point :math:`C` ", ""