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




Geometry
---------

The structure is composed of a full rotor of length :math:`L` with a constant circular cross section with a disk located at a third of its length.


.. image:: images/100002000000020E00000120659BD39C6FB9B062.png
    :width: 4.0547in
    :height: 1.9217in


.. _RefImage_100002000000020E00000120659BD39C6FB9B062.png:

**Modeling:**


.. csv-table::

    "", "Mass (:math:`\mathit{kg}`)", ":math:`{I}_{\mathit{zz}}` (:math:`{\mathit{kg.m}}^{2}`)", ":math:`{I}_{\mathit{xx}}\mathrm{=}{I}_{\mathit{yy}}` (:math:`{\mathit{kg.m}}^{2}`)"
    "Disk", "16.47", "0.1861", "0.09427"


**Table** 1.1-1 **: Disc characteristics**

**Beam length:**

:math:`L=0.4m`

**Circular section:**

Diameter: :math:`D=0.02m`




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

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

:math:`\nu =0.3`

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




Boundary conditions
----------------------

The rotor is supported by two infinitely rigid bearings at both ends and on an elastic bearing with viscous damping at two-thirds of its length. The tier coefficients are as follows:

:math:`{K}_{\mathit{xx}}={2.10}^{5}{\mathit{N.m}}^{-1}`; :math:`{K}_{\mathit{yy}}={5.10}^{5}{\mathit{N.m}}^{-1}`; :math:`{K}_{\mathit{xy}}={K}_{\mathit{yx}}=\mathrm{0 }`

:math:`{C}_{\mathit{xx}}={4.10}^{1}N/({\mathit{m.s}}^{-1})`; :math:`{C}_{\mathit{yy}}\mathrm{=}{1.10}^{2}N\mathrm{/}({\mathit{m.s}}^{\mathrm{-}1})`; :math:`{C}_{\mathit{xy}}={C}_{\mathit{yx}}=0`