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 disks, and 4 shaft sections.


.. image:: images/10000200000003EE0000016F16596A1FEEB720A8.png
    :width: 5.9811in
    :height: 1.7602in


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**Figure** 1.1-a **-a: Rotor model with 3 disks and 2 asymmetric bearings**


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}=200.{10}^{-3}\mathrm{kg}{m}^{2}`|:math:`{I}_{P}={400.10}^{-3}\mathrm{kg}{m}^{2}`|
+        +-------------------+------------------------------+-----------------------------------------------+-----------------------------------------------+
|        |:math:`\mathrm{D2}`|:math:`M=17\mathrm{kg}`       |:math:`{I}_{D}=170.{10}^{-3}\mathrm{kg}{m}^{2}`|:math:`{I}_{P}={340.10}^{-3}\mathrm{kg}{m}^{2}`|
+        +-------------------+------------------------------+-----------------------------------------------+-----------------------------------------------+
|        |:math:`\mathrm{D3}`|:math:`M=10\mathrm{kg}`       |:math:`{I}_{D}=15.{10}^{-3}\mathrm{kg}{m}^{2}` |:math:`{I}_{P}=30.{10}^{-3}\mathrm{kg}{m}^{2}` |
+--------+-------------------+------------------------------+-----------------------------------------------+-----------------------------------------------+

The characteristics of the levels are given in the tables that follow.

:math:`\Omega =0\mathrm{tr}/\mathrm{min}`

.. csv-table::

    "Level", ":math:`\mathrm{P1}` "," :math:`{K}_{\mathrm{yy}}={90.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{50.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{8.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{9.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=15.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}45.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}1.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{1.10}^{2}\mathit{Ns}\mathrm{/}m`"
    "", "", "", ""
    "Level", ":math:`\mathrm{P2}` "," :math:`{K}_{\mathrm{yy}}={60.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{15.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{8.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{8.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=12.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}19.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}1.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{1.10}^{2}\mathit{Ns}\mathrm{/}m`"


:math:`\Omega =5000\mathrm{tr}/\mathrm{min}`

.. csv-table::

    "Level", ":math:`\mathrm{P1}` "," :math:`{K}_{\mathrm{yy}}={90.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{50.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{9.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{9.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=15.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}45.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}1.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{1.10}^{2}\mathit{Ns}\mathrm{/}m`"
    "", "", "", ""
    "Level", ":math:`\mathrm{P2}` "," :math:`{K}_{\mathrm{yy}}={60.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{15.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{8.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{8.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=12.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}19.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}1.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{1.10}^{2}\mathit{Ns}\mathrm{/}m`"


:math:`\Omega =6500\mathrm{tr}/\mathrm{min}`

.. csv-table::

    "Level", ":math:`\mathrm{P1}` "," :math:`{K}_{\mathrm{yy}}={100.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{40.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{15.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{15.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=13.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}33.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}1.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{1.10}^{2}\mathit{Ns}\mathrm{/}m`"
    "", "", "", ""
    "Level", ":math:`\mathrm{P2}` "," :math:`{K}_{\mathrm{yy}}={70.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{14.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{13.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{13.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=10.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}15.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}1.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{1.10}^{2}\mathit{Ns}\mathrm{/}m`"


:math:`\Omega =8000\mathrm{tr}/\mathrm{min}`

.. csv-table::

    "Level", ":math:`\mathrm{P1}` "," :math:`{K}_{\mathrm{yy}}={110.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{35.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{20.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{20.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=11.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}26.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}2.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{2.10}^{2}\mathit{Ns}\mathrm{/}m`"
    "", "", "", ""
    "Level", ":math:`\mathrm{P2}` "," :math:`{K}_{\mathrm{yy}}={80.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{14.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{20.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{20.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=9.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}13.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}2.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{2.10}^{2}\mathit{Ns}\mathrm{/}m`"


:math:`\Omega =10000\mathrm{tr}/\mathrm{min}`

.. csv-table::

    "Level", ":math:`\mathrm{P1}` "," :math:`{K}_{\mathrm{yy}}={115.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{33.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{35.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{35.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=10.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}20.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}2.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{3.10}^{2}\mathit{Ns}\mathrm{/}m`"
    "", "", "", ""
    "Level", ":math:`\mathrm{P2}` "," :math:`{K}_{\mathrm{yy}}={90.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{14.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{30.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{30.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=8.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}10.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}2.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{2.10}^{2}\mathit{Ns}\mathrm{/}m`"


:math:`\Omega =14000\mathrm{tr}/\mathrm{min}`

.. csv-table::

    "Level", ":math:`\mathrm{P1}` "," :math:`{K}_{\mathrm{yy}}={120.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{30.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{70.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{70.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=7.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}15.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}3.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{4.10}^{2}\mathit{Ns}\mathrm{/}m`"
    "", "", "", ""
    "Level", ":math:`\mathrm{P2}` "," :math:`{K}_{\mathrm{yy}}={100.10}^{6}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{14.10}^{7}N\mathrm{/}m`"
    "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}\mathrm{-}{60.10}^{4}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{60.10}^{4}N\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathrm{yy}}=6.{10}^{4}\mathrm{Ns}/m` "," :math:`{c}_{\mathit{xx}}\mathrm{=}8.{10}^{4}\mathit{Ns}\mathrm{/}m`"
    "", "", ":math:`{c}_{\mathit{yx}}\mathrm{=}3.{10}^{2}\mathit{Ns}\mathrm{/}m` "," :math:`{c}_{\mathit{xy}}\mathrm{=}\mathrm{-}{3.10}^{2}\mathit{Ns}\mathrm{/}m`"



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

To block rigid body movements in the :math:`z` direction, we block the degrees of freedom :math:`\mathit{DZ}` and :math:`\mathit{DRZ}` at the level node :math:`\mathrm{B1}`.