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


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

A direct modal calculation is used to obtain the natural frequencies modified by the fluid around the spheres.

The two spheres are deduced by a vector translation of components :math:`(\mathrm{2,}0.,0.)` and for a rotation of nautical analyses :math:`(90.,0.,90.)`, which makes it possible to consider building a generalized model on the 1st sphere. It is possible to perform the same modal calculation with fluid/structure coupling using dynamic substructuring. The **direct** calculation taking into account the added mass leads to the following natural frequencies of the submerged system:


.. csv-table::

    ":math:`{f}_{1}=` "," :math:`3.1172\mathrm{Hz}`"
    ":math:`{f}_{2}=` "," :math:`3.1183\mathrm{Hz}`"
    ":math:`{f}_{3}=` "," :math:`9.2727\mathrm{Hz}`"
    ":math:`{f}_{4}=` "," :math:`9.8267\mathrm{Hz}`"
    ":math:`{f}_{5}=` "," :math:`22.4400\mathrm{Hz}`"
    ":math:`{f}_{6}=` "," :math:`30.3295\mathrm{Hz}`"



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

Direct modal calculation by*Code_Aster*.


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

The uncertainties in the solution are linked to the discretization of the fluid/structure interface and to the invariance of the mesh by rotation of nautical angles :math:`(90°,0°,90°)`.