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 2 discs and 3 shaft sections.


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.. image:: images/Object_9.svg
    :width: 579
    :height: 214


.. _RefImage_Object_9.svg:

Image 1.1-1: Rotor model with 2 discs 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=3.5\mathrm{kg}`      |:math:`{I}_{D}=3.5{10}^{-3}\mathrm{kg}{m}^{2}`|:math:`{I}_{P}={7.10}^{-3}\mathrm{kg}{m}^{2}`|
+--------+-------------------+------------------------------+----------------------------------------------+---------------------------------------------+
|        |:math:`\mathrm{D2}`|:math:`M=3.0\mathrm{kg}`      |:math:`{I}_{D}=3.0{10}^{-3}\mathrm{kg}{m}^{2}`|:math:`{I}_{P}={6.10}^{-3}\mathrm{kg}{m}^{2}`|
+--------+-------------------+------------------------------+----------------------------------------------+---------------------------------------------+

**Table** 1.2-1 **: Geometric characteristics and material**

The characteristics of the bearings are indicated in the following table.


.. csv-table::

    "Level", ":math:`\mathrm{P1}` "," :math:`{K}_{\mathrm{xx}}={8.10}^{7}N/m` "," :math:`{K}_{\mathrm{zz}}={1.10}^{8}N/m`"
    "", "", ":math:`{K}_{\mathrm{xz}}=-{1.10}^{7}N/m` "," :math:`{K}_{\mathrm{zx}}=-{6.10}^{7}N/m`"
    "", "", ":math:`{C}_{\mathrm{xx}}=8.{10}^{3}\mathrm{Ns}/m` "," :math:`{C}_{\mathrm{zz}}=1.2{10}^{4}\mathrm{Ns}/m`"
    "", "", ":math:`{C}_{\mathrm{xz}}=-3.{10}^{3}\mathrm{Ns}/m` "," :math:`{C}_{\mathrm{zx}}=-{3.10}^{3}\mathrm{Ns}/m`"
    "", "", "", ""
    "Level", ":math:`\mathrm{P2}` "," :math:`{K}_{\mathrm{xx}}=5.{10}^{7}N/m` "," :math:`{K}_{\mathrm{zz}}=7.{10}^{7}N/m`"
    "", "", ":math:`{K}_{\mathrm{xz}}=-2.{10}^{6}N/m` "," :math:`{K}_{\mathrm{zx}}=-4.{10}^{7}N/m`"
    "", "", ":math:`{C}_{\mathrm{xx}}=6.{10}^{3}\mathrm{Ns}/m` "," :math:`{C}_{\mathrm{zz}}=8.{10}^{3}\mathrm{Ns}/m`"
    "", "", ":math:`{C}_{\mathrm{xz}}=-1.5{10}^{3}\mathrm{Ns}/m` "," :math:`{C}_{\mathrm{zx}}=-1.5{10}^{3}\mathrm{Ns}/m`"


Table 1.2-2: Characteristics of bearings




Boundary conditions and loads
-------------------------------------

To block rigid body movements in the :math:`y` direction, we block the degrees of freedom :math:`\mathrm{DY}` and :math:`\mathrm{DRY}` at the level node :math:`\mathrm{B1}`.

Charging is a harmonic force at a constant speed of rotation.