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


Geometry
---------

The system is composed of a set of 4 springs, 2 point masses, supported by 3 supports.


.. image:: images/10000200000002AC0000008470BF50AA627A189B.png
    :width: 6.889in
    :height: 1.3311in


.. _RefImage_10000200000002AC0000008470BF50AA627A189B.png:


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

Link stiffness: :math:`k={k}_{\mathrm{1 }}={k}_{\mathrm{2 }}={10}^{\mathrm{3 }}N/m` :math:`{k}_{\mathrm{3 }}={k}_{\mathrm{4 }}=2k={2.10}^{\mathrm{3 }}N/m`;

point mass: :math:`m={m}_{\mathrm{1 }}={m}_{\mathrm{2 }}=\mathrm{10 }\mathrm{kg}`.


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

**Boundary conditions**:

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

Points :math:`\mathrm{NO1}`, :math:`\mathrm{NO3}`, and :math:`\mathrm{NO5}` 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`.

**Loading**:

The structure is subject to multiple spectral seismic excitation and to differential displacements:


* The pseudo-acceleration oscillator response spectra are simplified. Only the values corresponding to the 2 natural frequencies of the system are mentioned. They do not depend on depreciation:

     at node :math:`\mathit{NO}1`:

:math:`{\mathit{SRO}}_{\mathit{NO}1}({f}_{1})={A}_{11}=7m/{s}^{2}`

:math:`{\mathit{SRO}}_{\mathit{NO}1}({f}_{2})={A}_{21}=5m/{s}^{2}`

     at node :math:`\mathit{NO}3`:

:math:`{\mathit{SRO}}_{\mathit{NO}3}({f}_{1})={A}_{12}=7.7m/{s}^{2}`

:math:`{\mathit{SRO}}_{\mathit{NO}3}({f}_{2})={A}_{22}=5.5m/{s}^{2}`

     at node :math:`\mathit{NO}5`:

:math:`{\mathit{SRO}}_{\mathit{NO}5}({f}_{1})={A}_{13}=12m/{s}^{2}`

:math:`{\mathit{SRO}}_{\mathit{NO}5}({f}_{2})={A}_{23}=6m/{s}^{2}`

The excitations at nodes :math:`\mathit{NO}1` and :math:`\mathit{NO}3` are correlated. Two uncorrelated groups of supports are formed: group 1 is composed of nodes :math:`\mathit{NO}1` and :math:`\mathit{NO}3`; group 2 consists of the only node :math:`\mathit{NO}5`. The same multiple spectral seismic excitation is identical for the two models A and B.


* The differential movements are as follows:

Modeling A:

at node :math:`\mathit{NO}1`:

:math:`{\mathit{DDS}}_{\mathit{NO}1}={D}_{1}=-0.04m`

at node :math:`\mathit{NO}3`:

:math:`{\mathit{DDS}}_{\mathit{NO}3}={D}_{2}=-0.044m`

at node :math:`\mathit{NO}5`:

:math:`{\mathit{DDS}}_{\mathit{NO}5}={D}_{3}=0.06m`

Modeling B:

to support group 1 (DDS intra-group equals):


* :math:`{\mathit{DDS}}_{\mathit{NO}1}={\mathit{DDS}}_{\mathit{NO}3}={D}_{1}=-0.04m`

to support group 2:


* :math:`{\mathit{DDS}}_{\mathit{NO}5}={D}_{2}=0.06m`


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

The system is at rest.