4. Summary of results

FIGS. 4a and 4b represent the analytical radial displacement as a function of frequency compared to those obtained by the calculation at points \(A\), \(B\) and \(C\) (positioned in FIG. 1.3-a above) for a radius of \(7m\) outside the structural domain by post-processing with*Code_Aster*.

There is a good agreement between the results within the set. In particular, we find the resonance frequencies quite well in general around \(19\mathrm{Hz}\) and \(25\mathrm{Hz}\) despite a slight shift and the appearance of out-of-resonance movements at points A and C in the equatorial and vertical planes. However, there is a significant difference in resonance amplitudes linked at least to the previous slight lag. On the other hand, at least one other disturbance is observed with a parasitic resonance around \(10\mathrm{Hz}\). This frequency is the first natural frequency of the sphere with mass of water added to the center that can be found by a modal analysis calculation. The corresponding mode is an incompressible fluid mode different from the swelling modes sought here. This disturbance can be attenuated by introducing the parameter RFIC (here having the value of 0.5) as the value of MISS3D.

The use of static modes constrained on the outer envelope of the sphere is therefore exhaustive in order to represent swelling modes with spherical symmetry, but it is therefore also likely to represent modes of another nature that may disturb the solution, in particular at point \(B\), in particular at point where limit conditions have not been explicitly imposed in order to regain this spherical symmetry of the movements. The introduction of structural damping would also be likely to reduce the amplitude differences at resonance frequencies between the analytical results and the calculation.

Figure 4a: analytical and calculated displacements of the finite dimensional sphere test (freq < 15Hz)

Figure 4b: analytical and calculated displacements of the finite dimensional sphere test (freq > 15Hz)