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


Geometry
---------


.. image:: images/Object_1.svg
    :width: 452
    :height: 249


.. _RefImage_Object_1.svg:


.. csv-table::

    "Beam length", ":math:`L=20.0m`"
    "Inner diameter", ":math:`d=0.388m`"
    "Outside diameter", ":math:`D=0.400m`"
    "Mass concentrated at the top", ":math:`M=300.\mathrm{kg}`"
    "Mass moment of inertia", ":math:`J=200.\mathrm{kg}{m}^{2}`"



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


.. csv-table::

    "Tube density", ":math:`\rho =7850{\mathrm{kg.m}}^{-3}`"
    "Young's module", ":math:`E=210.E+9{\mathrm{N.m}}^{-2}`"
    "Poisson's Ratio", ":math:`\nu =0.`"



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

The tube is embedded at the base. The mass is free. Movement is allowed in a :math:`(\mathrm{DX},\mathrm{DRZ})` vertical plane.

A random effort :math:`F(t)`, applied to the concentrated mass, is assimilated to a stationary random Gaussian, centered process, of the white noise type with a limited band from :math:`\mathrm{1.Hz}` to :math:`\mathrm{101.Hz}`. It is characterized by a standard deviation :math:`{\sigma }_{F}=1\mathrm{kN}`, and a one-sided spectral density in frequency :math:`{S}_{F}(f)` such that:

:math:`\begin{array}{}\forall f\in [1\mathrm{Hz},101z]\\ {S}_{F}(f)=\frac{{\sigma }_{F}^{2}}{100}={10}^{4}{N}^{2}s\end{array}`


.. image:: images/Object_6.svg
    :width: 452
    :height: 249


.. _RefImage_Object_6.svg: