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


Geometry
---------

The structure is a rigid skate modelled by a single node. The skate is equipped with a return spring. It is subject to an external harmonic force :math:`{F}_{\mathit{ext}}(t)\mathrm{=}a\mathrm{sin}(2\pi ft)`. It carries a condition of normal shock against a rigid plan, treated by penalization.


.. image:: images/100000000000031A00000154C7065C0563DE8DA4.tif
    :width: 3.8228in
    :height: 1.3937in


.. _RefImage_100000000000031A00000154C7065C0563DE8DA4.tif:

**Figure** 1.1-a **: Geometry**

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

Properties of the structure:

:math:`m=156\mathrm{kg}`

:math:`{k}_{\mathit{rappel}}\mathrm{=}2\mathrm{\cdot }{10}^{6}N\mathrm{/}m`

Properties of the rubber bump stop:

:math:`{k}_{\mathrm{choc}}={10}^{10}N/m`

:math:`{c}_{\mathrm{choc}}=0\mathrm{Ns}/m`

:math:`{\mathrm{jeu}}_{\mathrm{choc}}=1\mathrm{mm}`


Initial conditions, limits and loading
------------------------------------------------

The skate starts with zero initial conditions: :math:`{x}_{0}={x}_{t=0}=0` and :math:`\dot{x}{}_{0}=\dot{x}{}_{t=0}=0`.

It only moves in one direction.

The external force is sinusoidal: :math:`{F}_{\mathrm{ext}}(t)=\mathrm{Fa}\cdot \mathrm{sin}(2\pi ft)`, with :math:`\mathrm{Fa}=3\cdot {10}^{3}N` and :math:`f=5\mathrm{Hz}`.