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


Calculation method
-----------------

The reference solution adopted is that obtained by solving the vibrating string equation:


.. image:: images/1000000000000207000001DBAF1FE05E64E73BB7.png
    :width: 1.9689in
    :height: 1.8035in


.. _RefImage_1000000000000207000001DBAF1FE05E64E73BB7.png:

The equation for vibrating strings is:

:math:`\frac{{\partial }^{2}y}{\partial {t}^{2}}=\frac{{F}_{0}}{\rho S}\frac{{\partial }^{2}y}{\partial {x}^{2}}`

with:

:math:`{F}_{0}`: tension in the rope,

:math:`\rho`: density,

:math:`S`: rope section.

For a string of length :math:`L`, fixed at these 2 ends, the nth natural vibration frequency is:

:math:`f=\frac{n}{2L}\sqrt{\frac{{F}_{0}}{\rho S}}`


Reference quantities and results
-----------------------------------

The quantities tested are the natural vibration frequencies of the cables for 2 calculation moments, which correspond to 2 different voltages.


.. csv-table::

    "**Not**", "**Cable Voltage**", "**Frequency**"
    "10"," :math:`6171.050459855N` "," :math:`62.73227096292042\mathit{Hz}`"
    "20"," :math:`12345.2954376N` "," :math:`88.72830898253936\mathit{Hz}`"



Uncertainties about the solution
----------------------------

The solution of the vibrating string equation is obtained under the following hypotheses:


* The movements of the rope around its equilibrium position remain small.


* the variations in angle :math:`\alpha` remain small.


* the variation in the tension of the moving rope remains small.

Under these assumptions, the reference solution is free of uncertainty.