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


Geometry
---------

The structure is modelled by a set of 3 springs and 2 point masses.


.. image:: images/10000000000002740000009E15092525028A2B97.png
    :width: 4.8681in
    :height: 1.2543in


.. _RefImage_10000000000002740000009E15092525028A2B97.png:


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


* Link stiffness: :math:`{k}_{1}={k}_{3}=k=\mathrm{100000 }N/m`; :math:`{k}_{2}=2k=200000N/m`


* Point mass: :math:`{m}_{2}={m}_{3}=m=2533\mathrm{kg}`.


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


* **boundary conditions**

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

Points :math:`\mathrm{NO1}` and :math:`\mathrm{NO4}` are embedded:

 :math:`\mathrm{DX}=\mathrm{DY}=\mathrm{DZ}=\mathrm{DRX}=\mathrm{DRY}=\mathrm{DRZ}=0`.

The other points are free to translate in the direction :math:`x`:

:math:`\mathrm{DY}=\mathrm{DZ}=\mathrm{DRX}=\mathrm{DRY}=\mathrm{DRZ}=0`.


* **load**

*Modeling A*: the structure is subjected to a decorrelated multiple spectral seismic excitation.

The pseudo-acceleration oscillator response spectra are defined by:


* at node :math:`\mathrm{NO1}`: :math:`{\mathrm{SRO}}_{\mathrm{NO1}}=\frac{{a}_{1}{\omega }^{2}}{∣{\omega }_{1}^{2}-{\omega }^{2}∣}`


* at node :math:`\mathrm{NO4}`: :math:`{\mathrm{SRO}}_{\mathrm{NO4}}=\frac{{a}_{2}{\omega }^{2}}{∣{\omega }_{2}^{2}-{\omega }^{2}∣}`

with :math:`{\omega }_{1}=2\pi {f}_{1}` :math:`{\omega }_{2}=2\pi {f}_{2}`

:math:`{f}_{1}=1.5\mathrm{Hz}`, :math:`{f}_{2}=2.\mathrm{Hz}`, :math:`{a}_{1}={a}_{2}=0.5{\mathrm{ms}}^{\text{-2}}`

They do not depend on depreciation.

*B* modeling: the structure is subjected to seismic excitation identical to both supports.

The pseudo-acceleration oscillator response spectrum is defined by:


* at node :math:`\mathrm{NO1}` and at node :math:`\mathrm{NO4}`: :math:`\mathrm{SRO}=\frac{{a}_{1}{\omega }^{2}}{∣{\omega }_{1}^{2}-{\omega }^{2}∣}`

             with :math:`{\omega }_{1}=2\pi {f}_{1}`

:math:`{f}_{1}=1.5\mathrm{Hz}`, :math:`{a}_{1}=0.5{\mathrm{ms}}^{\text{-2}}`

 It does not depend on depreciation.


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

The system is initially at rest