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


Geometry
---------

We calculate the response of a linear system composed of two masses and two independent springs to an acceleration imposed at their anchor point :math:`{P}_{1}`:


.. image:: images/10000201000000E7000000E9536EB9CCA0C3DA44.png
    :width: 2.4063in
    :height: 2.4272in


.. _RefImage_10000201000000E7000000E9536EB9CCA0C3DA44.png:

We have :math:`{P}_{1}{P}_{2}={P}_{1}{P}_{3}=1m` distance.


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


* connection stiffness: :math:`{k}_{1}=100N/m`;


* point masses: :math:`m=m({P}_{2})=m({P}_{3})=100\mathit{kg}`.


* :math:`\text{5 \%}` modal damping for all modes


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

The unknowns of displacement :math:`u`, speed :math:`\dot{u}` and accelerations :math:`\ddot{u}` that are discussed in this document are defined in the coordinate system relating to the training movement.

**Boundary conditions**

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

In the coordinate system relating to the imposed displacement, the point :math:`{P}_{1}` is embedded: :math:`{u}_{x}={u}_{y}={u}_{z}=0`.

**Loading**

In the global frame of reference, the anchor point :math:`{P}_{1}` is subject to an imposed harmonic displacement of frequency :math:`{f}_{\mathrm{ex}}`. The calculation is done from :math:`0` to :math:`20s` in the associated relative coordinate system.


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

The system is initially assumed to be at rest in the relative coordinate system: at :math:`t=0`, :math:`{u}_{x}(0)=0` and :math:`\dot{{u}_{x}}(0)=0` at all points.