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

Geometry
---------

The seismic response of a mass-spring system with a degree of freedom that can impact a fixed wall (problem 1) is compared to that of two identical mass-spring systems that can collide and subject to the same seismic stress (problem 2).


+----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                                                                                  |                                                                                             |
+ .. image:: images/Object_1.svg                                                   + .. image:: images/Object_2.svg                                                              +
|    :width: 226                                                                   |    :width: 2.8043in                                                                         |
+    :height: 96                                                                   +    :height: 0.9874in                                                                        +
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+                                                                                  +                                                                                             +
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+----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|**Problem 1**                                                                     |**Problem 2**                                                                                |
+----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------+



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

Spring stiffness: :math:`k=98696N/m`.

Point mass: :math:`m=25\mathit{kg}`.

For problem 1 (impact on a rigid wall), the normal shock stiffness is :math:`{K}_{\mathrm{choc}}=\mathrm{5,76}{10}^{7}N/m`. As for problem 2 (shock of two deformable structures), it is worth :math:`{K}_{\mathrm{choc}}=\mathrm{2,88}{10}^{7}N/m`.

In both cases, shock absorption is zero.

**For G modeling:**

Spring stiffness: :math:`k=500N/m`.

Point mass: :math:`m=15\mathit{kg}`.

The table above gives the characteristics of material DIS_CHOC_ENDO.

The stiffness under discharge is constant, the damping is zero.


.. csv-table::

    "Ux
    [m]", "Strength
    [N]", "Stiffness
    [N/m]", "Damping
    [N.s/m]"
    "0.00", "0.0", "2000.0", "0.0"
    "0.20", "400.0", "2000.0", "0.0"
    "0.50", "450.0", "2000.0", "0.0"
    "0.70", "400.0", "2000.0", "0.0"
    "0.95", "375.0", "2000.0", "0.0"
    "1.30", "350.0", "2000.0", "0.0"
    "1.60", "300.0", "2000.0", "0.0"
    "20.0", "300.0", "2000.0", "0.0"


The figure below shows the behavior corresponding to the data.


.. image:: images/1000020100000CC6000006654DFAFC3B70645267.png
    :width: 5.9055in
    :height: 2.9484in


.. _RefImage_1000020100000CC6000006654DFAFC3B70645267.png:


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

**Boundary conditions**

The only authorized movements are translations according to axis :math:`x`.

Points :math:`A`, :math:`B`, and :math:`C` are embedded: :math:`\mathrm{dx}=\mathrm{dy}=\mathrm{dz}=0`.

**Loading**

The anchor points :math:`A` and :math:`B` are subject to acceleration in the direction :math:`x`. Point C is subject to the opposite acceleration.


* A to F models: :math:`{\gamma }_{1}(t)=\mathrm{sin}\omega t` with :math:`\omega =20.\pi {s}^{-1}` (:math:`{\gamma }_{2}(t)=-\mathrm{sin}\omega t` for point C).


* G modeling: :math:`{\gamma }_{1}(t)=\alpha t\mathrm{sin}\omega t` with :math:`\omega =20.\pi {s}^{-1}` and :math:`\alpha =1E5`.


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

In both cases, the mass-spring systems are initially at rest:

to :math:`t=0`, :math:`\mathrm{dx}(0)=0`, :math:`\mathrm{dx}/\mathrm{dt}(0)=0` in every way.

For problem 1, the mass is separated from the fixed wall in game :math:`j=5.{10}^{-4}m`. As for problem 2, the masses are separate from game :math:`J=2j={10}^{-3}m`.