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


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

A detailed analytical solution is available in reference [:ref:`bib1 <bib1>`].

The following notations are adopted:


.. csv-table::

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


We also ask:

:math:`{\omega }_{n}=(\mathrm{2n}-1)\frac{\pi }{2}` where :math:`n=\mathrm{1,2}\mathrm{,3},\mathrm{...}`

:math:`{\varepsilon }_{n}=\frac{1}{2}(\alpha {\omega }_{n}+\beta /{\omega }_{n})`

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

:math:`u(x,t)=\frac{\mathrm{Nx}}{\mathrm{EA}}+\frac{\mathrm{8NL}}{{\pi }^{2}\mathrm{EA}}\underset{n=1}{\overset{\infty }{\Sigma }}{(-1)}^{n}\frac{{e}^{-{\omega }_{n}{\varepsilon }_{n}t}}{{(\mathrm{2n}-1)}^{2}}\left\{\mathrm{cos}(\sqrt{1-{\varepsilon }_{n}^{2}}{\omega }_{n}t)+\frac{{\varepsilon }_{n}}{\sqrt{1-{\varepsilon }_{n}^{2}}}\mathrm{sin}(\sqrt{1-{\varepsilon }_{n}^{2}}{\omega }_{n}t)\right\}`


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

The values of the displacement fields, speed and acceleration fields of the free end (node :math:`\mathrm{N10}`) are valid at time :math:`t=0.0195s`:


.. csv-table::

    "", "**Movement** :math:`(m)` ", "**Speed** :math:`(\mathrm{m.}{s}^{-1})` ", "**Acceleration** :math:`(\mathrm{m.}{s}^{-2})`"
    "Calculation without depreciation", "—8.3766x10—7", "1.6753 x 10—3", "0"
    "Calculation with structural depreciation", "—1.00462x10—6", "1.20384x10—3", "—1.21564"



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

Analytical solution.


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


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