2. Reference solution

2.1. Calculation method used for the reference solution

We are based on an analytical result [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 \(({S}_{1})\) is equal to:

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

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

\(f=\frac{1}{2\pi }\sqrt{\frac{\mathrm{2K}}{m+{m}_{a}}}\)

Digital application:

\(K={10}^{5}N/m\)

\(m=12\mathrm{kg}\)

\({m}_{a}=329.17\mathrm{kg}\)

\(F=3.8534\mathrm{Hz}\)

2.2. Bibliographical references

1. R.D. BLEVINS, « Formulas for Natural Frequency and Mode Shape, » Ed. KRIEGER