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


Geometry
---------

A simple rotor model supported by 2 bearings (first and last rotor nodes respectively), is composed of 3 disks, the shaft section is :math:`0.05m` in radius. It measures :math:`1.3m` (see figure below). The rotor is rotating along the Z axis.


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.. image:: images/Object_1.svg
    :width: 454
    :height: 217


.. _RefImage_Object_1.svg:

**Figure** 1.1-a-a: Rotor model with 3 discs from [:ref:`bib1 <bib1>`]

The respective lengths are:

:math:`{L}_{1}=0.2m` :math:`{L}_{2}=0.3m` :math:`{L}_{3}=0.5m` :math:`{L}_{4}=0.3m`


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

The characteristics of the material of the shaft and the discs are:


* Young's module :math:`E={2.10}^{11}`


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


* Poisson's ratio :math:`\nu =0.3`

The characteristics of the discs are:


.. csv-table::

    "Disc", ":math:`{D}_{1}` "," :math:`{D}_{2}` "," :math:`{D}_{3}`"
    "Thickness (:math:`m`)", "0.05", "0.05", "0.06"
    "Inner radius (:math:`m`)", "0.05", "0.05", "0.05"
    "Outside radius (:math:`m`)", "0.12", "0.12", "0.20", "0.20"


The characteristics of the bearings are:


.. csv-table::

    ":math:`{K}_{\mathit{xx}}\mathrm{=}{7.10}^{7}N\mathrm{/}m` "," :math:`{K}_{y}\mathrm{=}{5.10}^{7}N\mathrm{/}m` "," :math:`{K}_{\mathit{yx}}\mathrm{=}{K}_{\mathit{xy}}\mathrm{=}0`"
    ":math:`{C}_{\mathit{xx}}\mathrm{=}{7.10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{C}_{\mathit{yy}}\mathrm{=}{5.10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{C}_{\mathit{yx}}\mathrm{=}{C}_{\mathit{xy}}\mathrm{=}0`"



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

To block rigid body movements in the :math:`z` direction, we block the degree of freedom :math:`\mathit{DZ}` at the node of the first level (first node of the tree).