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


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

A detailed analytical solution is available in reference [:ref:`bib2 <bib2>`].

Let's use the following notations:


.. csv-table::

    ":math:`E` ", ":", "Young's modulus"
    ":math:`L` ", ":", "bar length"
    ":math:`A` ", ":", "bar section"
    ":math:`N` ", ":", "normal force directed along the :math:`X` axis"
    ":math:`\alpha ,\beta` ", ":", "Rayleigh damping coefficients"
    ":math:`\Omega` ", ":", "excitation frequency"


And let's put

:math:`r\mathrm{=}\sqrt{\frac{1+{\beta }^{2}\mathrm{/}{\Omega }^{2}}{1+{\alpha }^{2}\mathrm{/}{\Omega }^{2}}}`

:math:`k\mathrm{=}p+\mathit{iq}\mathrm{=}\Omega \sqrt{\frac{p}{2E}}\left[\sqrt{r\mathrm{-}\frac{1\mathrm{-}\alpha \beta }{1+{\alpha }^{2}{\Omega }^{2}}}+i\sqrt{r+\frac{1\mathrm{-}\alpha \beta }{1+{\alpha }^{2}{\Omega }^{2}}}\right]`

The displacement to any point :math:`M(x)` is given by:

:math:`V(x)\mathrm{=}\frac{N}{\mathit{EA}}\frac{A}{(p+\mathit{iq})(1+i\Omega \alpha )}\frac{\mathit{sh}\mathit{px}\mathrm{cos}\mathit{qx}+i\mathit{ch}\mathit{px}\mathrm{sin}\mathit{qx}}{\mathit{ch}L\mathrm{cos}\mathit{qL}+i\mathit{sh}\mathit{pL}\mathrm{sin}\mathit{qL}}`


.. csv-table::

    "", "**Movement** :math:`(m)` ", "**Speed** :math:`(m/s)` ", "**Acceleration** :math:`(m/{s}^{2})`"
    "Real game", "—7.00 10—11", "—3.18 10—6", "2.76 10—5"
    "Imaginary part", "5.07 10—9", "—4.40 10—8", "—2.00 10—3"



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

Fields of movement, speed, and acceleration of the free end of the bar.


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

Digital solution.


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


1. T. KERBER "Harmonic substructuring in *Code_Aster"*, Report EDF, HP‑61/93‑104.


2. G. ROBERT, Analytical solutions in structural dynamics, Samtech Report No. 121, March 1996.


3. P. RICHARD, Substructuring methods in *Code_Aster*, Internal Report EDF - DER, HP-61/92-149.