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


Geometry
---------


.. image:: images/Object_1.svg
    :width: 339
    :height: 148


.. _RefImage_Object_1.svg:

Characteristics:


.. csv-table::

    "Pendulum length", ":math:`L=0.6m`"
    "Eccentricity", ":math:`a=0.1m`"
    "Profile height", ":math:`h=0.01m`"
    "Profile width", ":math:`b=0.004m`"
    "Section", ":math:`S=\mathrm{bh}`"
    "Flexural inertia", ":math:`{I}_{Z}={\mathrm{bh}}^{3}/12`"



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


.. csv-table::

    "Young's module", ":math:`E=7.E10N/{m}^{2}`"
    "Poisson's Ratio", ":math:`\nu =0.3`"
    "Density", ":math:`\rho =2700\mathrm{kg}/{m}^{3}`"



Boundary conditions and loading
------------------------------------

The beam is hinged at point :math:`A`. The axis of the joint is axis :math:`Y`. The initial state of stress that makes it possible to calculate the geometric and centrifugal stiffness is obtained by imposing a speed of rotation and gravity.


.. csv-table::

    "Gravity acceleration", ":math:`g=-9.81m/{s}^{\mathrm{²}}` (parallel to the :math:`Z` axis)"
    "Rotation speed", ":math:`\Omega =10\mathrm{rad}/s`"


The static equilibrium position :math:`{\theta }_{0}` corresponding to the load is calculated by the relationship:

:math:`3g\mathrm{cos}{\theta }_{0}={\Omega }^{2}(\mathrm{3a}+\mathrm{2L}\mathrm{cos}{\theta }_{0})\mathrm{sin}{\theta }_{0}`

We find :math:`{\theta }_{0}=11.269931365°`

The conditions for travel to point :math:`A` are as follows:

:math:`u=v=w=0`; :math:`{\phi }_{X}={\phi }_{Z}=0`

It is also considered that the section through :math:`A` remains rigid.


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

Not applicable in modal analysis.