2. Benchmark solution#

2.1. 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 \((\mathrm{2,}0.,0.)\) and for a rotation of nautical analyses \((90.,0.,90.)\), 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:

\({f}_{1}=\)

\(3.1172\mathrm{Hz}\)

\({f}_{2}=\)

\(3.1183\mathrm{Hz}\)

\({f}_{3}=\)

\(9.2727\mathrm{Hz}\)

\({f}_{4}=\)

\(9.8267\mathrm{Hz}\)

\({f}_{5}=\)

\(22.4400\mathrm{Hz}\)

\({f}_{6}=\)

\(30.3295\mathrm{Hz}\)

2.2. Benchmark results#

Direct modal calculation by*Code_Aster*.

2.3. 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 \((90°,0°,90°)\).