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


Geometry
---------

The system is composed of a set of 2 springs, 3 point masses, supported by 2 supports:


.. image:: images/1000020000000FA0000008CAF35B383A6467FDCD.png
    :width: 4.3291in
    :height: 0.7717in


.. _RefImage_1000020000000FA0000008CAF35B383A6467FDCD.png:

**Figure** 1.1-1

     The system consists of the following elements:


* a central mcenter mass of 10.0 kg at point N3;


* two mleft and mright masses of 4.5 and 3.7 kg respectively, fixed through two elastic springs each with a kouter stiffness of 103 N/m.


* The system is unidirectional, the masses only "slide" in the X direction.



     The coordinates of the points shown in the figure above are:



.. csv-table::

    "Node", "X (m)", "Y (m)", "Z (m)"
    "N1", "0.0", "0.0", "0.0"
    "N2", "0.1", "0.0", "0.0"
    "N3", "0.2", "0.0", "0.0"
    "N4", "0.3", "0.0", "0.0"
    "N5", "0.4", "0.0", "0.0"


Table 1.1-1

The masses are linked together by various phenomena and devices characterized by non-linear behavioral relationships.


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

**Boundary conditions**:

All the knots in which the weights are carried are free only in the X direction: :math:`\mathit{dy}=\mathit{dz}=\mathit{drx}=\mathit{dry}=\mathit{drz}=0`

Nodes :math:`N1` and :math:`N2` are embedded: :math:`\mathit{dx}=\mathit{dy}=\mathit{dz}=\mathit{drx}=\mathit{dry}=\mathit{drz}=0`.

**Loading**:

The central mass is subjected to a time-based sine force (monofrequency) in the form: :math:`A\mathrm{sin}(2\mathrm{\pi }ft)`. The amplitude :math:`A` and the frequency :math:`f` differ according to the modeling in order to ensure that the loading level is sufficient to activate the non-linear relationship between masses. The loads for the various models are given in the following table:


.. csv-table::

    "Modeling", ":math:`A` (N)", ":math:`f` (Hz)"
    "A", "60.0", "2.0"
    "B", "500.0", "2.0"
    "C", "500.0", "2.0"
    "D", "500.0", "2.0"
    "E", "50.0", "2.0"
    "F", "10.0", "4.0"
    "G", "10.0", "4.0"


 **Table** 1.2-1

     The total loading time is 1.0 sec. The integration is carried out with a fixed step of 1.0E-6 sec.


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

The system is at rest.