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


Geometry
---------

The structure is composed of a rotor of length :math:`L` with a variable circular section with two infinitely rigid bearings at the level of the supports. The diameters of the rotor section vary linearly depending on the length.


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

The geometric and material characteristics are listed in the following table.


+----------------+-------------------------------------------------------------------------+-----------------------------+-------------------------------------+----------------+
|Material                                                                                  |:math:`E=2{10}^{11}N/{m}^{2}`|:math:`\rho =7800\mathrm{kg}/{m}^{3}`|:math:`\nu =0.3`|
+----------------+-------------------------------------------------------------------------+-----------------------------+-------------------------------------+----------------+
|Rotor length    |:math:`L=1m`                                                                                                                                                  |
+----------------+-------------------------------------------------------------------------+-----------------------------+-------------------------------------+----------------+
|Section diameter|original outer diameter: :math:`\mathrm{De1}=\mathrm{0,2}m`                                                                                                   |
+                +-------------------------------------------------------------------------+-----------------------------+-------------------------------------+----------------+
|                |final outer diameter: :math:`\mathrm{De2}=\mathrm{0,1}m`                                                                                                      |
+----------------+-------------------------------------------------------------------------+-----------------------------+-------------------------------------+----------------+
|Thickness       |at any point on the rotor: :math:`\mathrm{De}–\mathrm{Di}=\mathrm{0,04}m`                                                                                     |
+----------------+-------------------------------------------------------------------------+-----------------------------+-------------------------------------+----------------+

**Table** 1.2-1

The translation coefficients of the stages are:

:math:`\mathrm{Kxx}=\mathrm{Kyy}=1.0E+12{\mathrm{kg.s}}^{-2}`

:math:`\mathrm{Kxy}=\mathrm{Kyx}=0.0\mathrm{kg.s}-2`

:math:`\mathrm{Cxx}=\mathrm{Cyy}=\mathrm{Cxy}=\mathrm{Cyx}=0.0{\mathrm{kg.s}}^{-1}`

The torsional coefficients of the bearings are:

:math:`\mathrm{Krz}=\mathrm{Kry}=1.0E+12{\mathrm{kg.s}}^{-2}` :math:`\mathrm{Kxy}=\mathrm{Kyx}=0.0\mathrm{kg.s}-2` :math:`\mathrm{Crx}=\mathrm{Cry}=\mathrm{Cxy}=\mathrm{Cyx}=0.0{\mathrm{kg.s}}^{-1}`


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

The rotor is supported on two rigid bearings at both ends.