Reference solution
=====================


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

We are based on an analytical result [:ref:`bib1 <bib1>`]:

In the case of two concentric spheres immersed in the same fluid, it is shown that the added mass induced by the fluid confined to the internal sphere :math:`({S}_{1})` is equal to:

:math:`{m}_{a}=\frac{2}{3}{\rho }_{f}\pi \left[\frac{1+2{(\frac{{R}_{1}}{{R}_{2}})}^{3}}{1-{(\frac{{R}_{1}}{{R}_{2}})}^{3}}\right]{R}_{1}^{3}`

If we assume that the sphere has only one degree of freedom following :math:`\mathrm{Oz}`, the natural mode of translation of the sphere :math:`({S}_{1})` following :math:`\mathrm{Oz}` is given by:

:math:`f=\frac{1}{2\pi }\sqrt{\frac{\mathrm{2K}}{m+{m}_{a}}}`

Digital application:

.. csv-table::

    ":math:`K={10}^{5}N/m`"
    ":math:`m=12\mathrm{kg}`"
    ":math:`{m}_{a}=329.17\mathrm{kg}`"
    ":math:`F=3.8534\mathrm{Hz}`"



Bibliographical references
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1. R.D. BLEVINS, "Formulas for Natural Frequency and Mode Shape," Ed. KRIEGER