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

Geometry
---------


.. image:: images/Object_123.png
    :width: 2.9264in
    :height: 1.2457in


.. _RefSchema_Object_123.png:


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

*For models A and B*

:math:`K={10}^{-7}N/m`

:math:`M=1\mathrm{kg}`

The reaction force :math:`{F}_{\mathit{ext}}` depends on the compression of the wall which follows the law below:


.. image:: images/Object_135.png
    :width: 3.1709in
    :height: 2.1764in


.. _RefSchema_Object_135.png:

With:

:math:`{K}_{1}=1N/m`

:math:`{F}_{\mathit{fl}}=1N`

:math:`{F}_{\mathit{seuil}}=0.5N`

:math:`{K}_{2}=0.5N/m`

*For C* modeling

The force-compression relationship is shown in the figure below and in the details of the parameters.

With:

:math:`{K}_{1}=1N/m`

:math:`{F}_{\mathit{fl}}=1N`

:math:`{F}_{\mathit{seuil}}=0.5N`


.. image:: images/10000000000001E000000121C9B18786A48B64AC.png
    :width: 3.9366in
    :height: 2.3661in


.. _RefImage_10000000000001E000000121C9B18786A48B64AC.png:

:math:`\text{LARG\_PLAT}=1m`

:math:`\text{ENFO\_FL}=1m`

:math:`\text{DEPL\_POST\_FL}=[\mathrm{2.,}\mathrm{3.,}\mathrm{4.,}5.]m`

:math:`\text{RIGI\_POST\_FL}=[\mathrm{0.2,0}\mathrm{.15,0}\mathrm{.125,0}.1]N/m`

*For modelling* D

The force-compression relationship corresponds to that used in the C modeling, but damping is added. Before buckling this damping is equal to :math:`\text{AMOR\_NOR}`, then it is divided by two during buckling.

:math:`\text{AMOR\_NOR}=10.N/m\mathrm{.}{s}^{-1}`

:math:`\text{AMOR\_NOR}=5.N/m\mathrm{.}{s}^{-1}`

:math:`\text{DEPL\_POST\_FL}=[\mathrm{5.,}\mathrm{5.,}\mathrm{5.,}5.]N/m\mathrm{.}{s}^{-1}`


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

The movement of the mobile mass :math:`M` along the :math:`x` axis is imposed.

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

*For models A and B*

At the initial instant, the moving mass is just in contact against the wall (zero initial play) with an initial speed equal to :math:`2m/s`.

The wall has not yet undergone plastic deformation (wall not buckled).

*For the*modelling*s*C* and D

At the initial instant, the moving mass is just in contact against the wall (zero initial play) with an initial speed equal to :math:`2.828427125m/s`.

In the initial state, the wall has not yet undergone plastic deformation (wall not buckled).