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




Geometry
---------

A rotor model supported by 2 bearings (nodes :math:`\mathrm{B1}` and :math:`\mathrm{B2}` in the figure below), whose stiffness and damping matrices are non-symmetric. It is composed of 3 discs and 4 shaft sections.


.. image:: images/1000020000000548000001DA8FEF24624A1AB56B.png
    :width: 6.8717in
    :height: 2.4063in


.. _RefImage_1000020000000548000001DA8FEF24624A1AB56B.png:




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`                               |
+--------+-------------------+-----------------------------+------------------------------------------------+-----------------------------------------------+
|Disc    |:math:`\mathrm{D1}`|:math:`M=20\mathrm{kg}`      |:math:`{I}_{D}=\mathrm{0,20}\mathrm{kg}{m}^{2}` |:math:`{I}_{P}=\mathrm{0,40}\mathrm{kg}{m}^{2}`|
+--------+-------------------+-----------------------------+------------------------------------------------+-----------------------------------------------+
|        |:math:`\mathrm{D2}`|:math:`M=17\mathrm{kg}`      |:math:`{I}_{D}=\mathrm{0,17}\mathrm{kg}{m}^{2}` |:math:`{I}_{P}=\mathrm{0,34}\mathrm{kg}{m}^{2}`|
+--------+-------------------+-----------------------------+------------------------------------------------+-----------------------------------------------+
|        |:math:`\mathrm{D3}`|:math:`M=10\mathrm{kg}`      |:math:`{I}_{D}=\mathrm{0,015}\mathrm{kg}{m}^{2}`|:math:`{I}_{P}=\mathrm{0,30}\mathrm{kg}{m}^{2}`|
+--------+-------------------+-----------------------------+------------------------------------------------+-----------------------------------------------+

The characteristics of the bearings vary linearly according to the speed of rotation:


* to :math:`5000\mathrm{tr}/\mathrm{min}`:


.. csv-table::

    "Level", ":math:`\mathrm{P1}` "," :math:`\mathrm{kyy}=9{10}^{7}N/m` "," :math:`\mathrm{kzz}=5{10}^{8}N/m`"
    "", "", ":math:`\mathrm{kyz}=9{10}^{4}N/m` "," :math:`\mathrm{kzy}=-9{10}^{4}N/m`"
    "", "", ":math:`\mathrm{cyy}=\mathrm{1,5}{10}^{5}\mathrm{Ns}/m` "," :math:`\mathrm{czz}=4.5{10}^{5}\mathrm{Ns}/m`"
    "", "", ":math:`\mathrm{cyz}=-1{10}^{2}\mathrm{Ns}/m` "," :math:`\mathrm{czy}=1{10}^{2}\mathrm{Ns}/m`"
    "", "", "", ""
    "Level", ":math:`\mathrm{P2}` "," :math:`\mathrm{kyy}=6{10}^{7}N/m` "," :math:`\mathrm{kzz}=\mathrm{1,5}{10}^{8}N/m`"
    "", "", ":math:`\mathrm{kyz}=8{10}^{4}N/m` "," :math:`\mathrm{kzy}=-8{10}^{4}N/m`"
    "", "", ":math:`\mathrm{cyy}=\mathrm{1,2}{10}^{5}\mathrm{Ns}/m` "," :math:`\mathrm{czz}=\mathrm{1,9}{10}^{5}\mathrm{Ns}/m`"
    "", "", ":math:`\mathrm{cyz}=-1{10}^{2}\mathrm{Ns}/m` "," :math:`\mathrm{czy}=1{10}^{2}\mathrm{Ns}/m`"



* to :math:`6500\mathrm{tr}/\mathrm{min}`:


.. csv-table::

    "Level", ":math:`\mathrm{P1}` "," :math:`\mathrm{kyy}=1{10}^{8}N/m` "," :math:`\mathrm{kzz}=4{10}^{8}N/m`"
    "", "", ":math:`\mathrm{kyz}=\mathrm{1,5}{10}^{5}N/m` "," :math:`\mathrm{kzy}=-\mathrm{1,5}{10}^{5}N/m`"
    "", "", ":math:`\mathrm{cyy}=\mathrm{1,3}{10}^{5}\mathrm{Ns}/m` "," :math:`\mathrm{czz}=3.3{10}^{5}\mathrm{Ns}/m`"
    "", "", ":math:`\mathrm{cyz}=-1{10}^{2}\mathrm{Ns}/m` "," :math:`\mathrm{czy}=1{10}^{2}\mathrm{Ns}/m`"
    "", "", "", ""
    "Level", "P2"," :math:`\mathrm{kyy}=7{10}^{7}N/m` "," :math:`\mathrm{kzz}=\mathrm{1,4}{10}^{8}N/m`"
    "", "", ":math:`\mathrm{kyz}=\mathrm{1,3}{10}^{5}N/m` "," :math:`\mathrm{kzy}=-\mathrm{1,3}{10}^{5}N/m`"
    "", "", ":math:`\mathrm{cyy}=1{10}^{5}\mathrm{Ns}/m` "," :math:`\mathrm{czz}=\mathrm{1,5}{10}^{5}\mathrm{Ns}/m`"
    "", "", ":math:`\mathrm{cyz}=-1{10}^{2}\mathrm{Ns}/m` "," :math:`\mathrm{czy}=1{10}^{2}\mathrm{Ns}/m`"





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

To block rigid body movements in direction :math:`x`, we block the degree of freedom :math:`\mathrm{DX}` at the level node :math:`\mathrm{B1}`.