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


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

The period :math:`T` of a moving pendulum without friction around the fixed point :math:`O`, whose mass is concentrated at the center of gravity :math:`G` (:math:`\mathrm{OG}=l`) and whose maximum angular amplitude is :math:`{\theta }_{0}` is given by the series [:ref:`bib1 <bib1>`]:

:math:`T=2\pi \sqrt{\frac{l}{g}}\left[1+\underset{n=1}{\overset{\infty }{\Sigma }}{a}_{n}^{2}{(\mathrm{sin}\frac{{\theta }_{0}}{2})}^{\mathrm{2n}}\right]`

with

:math:`{a}_{n}=\frac{\mathrm{2n}-1}{\mathrm{2n}}`


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

For :math:`l=0.5m`, :math:`g=9.81m/{s}^{2}`, and :math:`{\theta }_{0}=\pi /2`, we get: :math:`T=1.6744s`


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

The terms of the series were summed up to and including :math:`n=12`, with the last term taken into account being less than :math:`{10}^{-5}` times the calculated sum.


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


1. J. HAAG, "Vibratory movements", P.U.F. (1952).