Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

The reference solution is the one given in sheet SDLL10 /89 of the guide VPCS, which presents the calculation method as follows:

Pulse :math:`{\omega }_{i}` is given by the roots of the equation:


.. image:: images/Object_6.svg
    :width: 297
    :height: 41


.. _RefImage_Object_6.svg:

with:


.. image:: images/Object_7.svg
    :width: 297
    :height: 41


.. _RefImage_Object_7.svg:

The translation components :math:`\nu` of the :math:`{F}_{i}(x)` mode are then:

:math:`{\phi }_{i}(x)\mathrm{=}{e}^{\alpha x}\left[\mathrm{cos}(\mathit{rx})\mathrm{-}\mathit{ch}(\mathit{rx})+\frac{\mathrm{cos}(\mathit{rL})\mathrm{-}\mathit{ch}(\mathit{sL})}{r\mathit{sh}(\mathit{sL})\mathrm{-}s\mathrm{sin}(\mathit{rL})}(s\mathrm{sin}(\mathit{rx})\mathrm{-}r\mathit{sh}(\mathit{sx}))\right]`


Benchmark results
----------------------

4 first natural frequencies and natural modes standardized to 1 for the largest translation component.


Uncertainty about the solution
---------------------------

Analytical solution.


Bibliographical references
---------------------------


* Dynamic Analysis Working Group. Structural Analysis Software Validation Committee. French Society of Mechanics (1988).