2. Benchmark solution#

2.1. Calculation method used for the reference solution#

For thin spheres (\(\mathrm{i.t}\ll R\) with \(i\), order of mode), the natural modes with radial and tangential displacement established by a membrane theory are given by [bib1] and [bib2]:

\({f}_{i}=\frac{{\lambda }_{i}}{2\pi R}\sqrt{\frac{E}{\rho (1-{\nu }^{2})}}\)

with \({\lambda }_{i}=\frac{1}{\sqrt{2}}\sqrt{b\pm \sqrt{{b}^{2}-4(1-{\nu }^{2})({i}^{2}+i-2)}}\) and \(b={i}^{2}+i+1+3\nu\)

The theory presented by Hayek makes it possible to introduce a correction of the flexure effect (approximation of the general Wilkinson theory) which leads to values of \({\lambda }_{i}\) as a function of

\(\begin{array}{}a={t}^{2}/12{R}^{2}\\ b=i(i+1)\end{array}\)

and solution of:

\({\lambda }_{i}^{4}-{\lambda }_{i}^{2}[1+3\nu -a(1-\nu )+b(1+a\nu +\mathrm{ab})]+\mathrm{ab}[{b}^{2}-\mathrm{4b}+5-{\nu }^{2}]+(1-{\nu }^{2})[b-2(1+a)]=0\)

2.2. Benchmark results#

Natural frequencies:

i

Natural frequencies

2

237.25

3

282.85

4

305.24

5

324.17

6

346.76

7

376.68

8

9

465.75

10

526.20

2.3. Uncertainty about the solution#

Analytical solution.

2.4. Bibliographical references#

  1. Sheet VPCS SDLS 07/89 in the Structural Analysis Software Validation Guide/ SFM AFNOR TECHNIQUE 1990.

    1. HAYEK: « Vibrations of a spherical shell in acoustic medium », Journal of the Acoustical Society of America, Vol. 40, 2, 1996, pp. 342-348