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




Geometry
---------

For purely demonstrative purposes, we study in this test case a "non-real" shaft line with a total length of 0.4 m and a full circular cross section with an external diameter of 0.02 m, supported by three levels. This shaft line has a disc positioned 130 mm from the left end.


.. image:: images/1000020000000FA0000008CA64B7F9D2C59AA60D.png
    :width: 5.7575in
    :height: 1.9972in


.. _RefImage_1000020000000FA0000008CA64B7F9D2C59AA60D.png:

**Figure** 1.1-1



     The coordinates of the points shown in the figure above are:



.. csv-table::

    "Node", "X (m)", "Y (m)", "Z (m)"
    "N1Z1", "0.000000E+00", "0.000000E+00", "0.000000E+00", "0.000000E+00"
    "N2Z1", "0.000000E+00", "0.000000E+00", "1.333340E-01"
    "N3Z1", "0.000000E+00", "0.000000E+00", "2.666670E-01"
    "N4Z1", "0.000000E+00", "0.000000E+00", "4.000000E+00", "4.000000E-01"
    "AP1 ", "1.000000E-05", "0.000000E+00", "0.000000E+00"
    "P1", "1.000000E-05", "0.000000E+00", "2.666670E-01"
    "AP2 ", "1.000000E-05", "0.000000E+00", "4.000000E-01"


Table 1.1-1




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

The material made of the shaft line has the following elastic characteristics:


* density: = 7800kg/m3


* Poisson's ratio: n = 0.3


* Young module: E = 200 GPa

Link stiffness modeling the bearing at the N1Z1 node: :math:`{k}_{x}={k}_{Y}={10}^{12}N/m`;

Link stiffness modeling the bearing at the N3Z1 node: :math:`{k}_{x}=2.0\times {10}^{5}N/m` and :math:`{k}_{Y}=5.0\times {10}^{5}N/m`;

Link stiffness modeling the bearing at the N4Z1 node: :math:`{k}_{x}={k}_{Y}={10}^{12}N/m`;

Rotor point mass: :math:`m=16.47\mathit{kg}`

The inertia tensor of the rotor with respect to the N2Z1 node are: :math:`{I}_{\mathit{XX}}=0.1861{\mathit{kgm}}^{2}`; :math:`{I}_{\mathit{YY}}={I}_{\mathit{ZZ}}=0.09427{\mathit{kgm}}^{2}` and the coupling terms and the eccentricity terms are zero.




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

**Boundary conditions**:

The nodes belonging to the tree line are blocked in terms of vertical movement as well as rotation around this axis: :math:`\mathit{dz}=\mathit{drz}=0`

Nodes :math:`\mathit{AP}1`, :math:`P1`, and :math:`\mathit{AP}2` are embedded: :math:`\mathit{dx}=\mathit{dy}=\mathit{dz}=\mathit{drx}=\mathit{dry}=\mathit{drz}=0`.

**Loading**:

The structure is subjected to an unbalance at point N2Z1 with the amplitude :math:`A=15` g.mm with a zero phase. The rotation speed is 2500 rpm.




Initial conditions
--------------------

The system is at rest.