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


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

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

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

with :math:`{\lambda }_{i}=\frac{1}{\sqrt{2}}\sqrt{b\pm \sqrt{{b}^{2}-4(1-{\nu }^{2})({i}^{2}+i-2)}}` and :math:`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 :math:`{\lambda }_{i}` as a function of

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

and solution of:

:math:`{\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`


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

Natural frequencies:


.. csv-table::

    "**i**", "**Natural frequencies**"
    "2", "237.25"
    "3", "282.85"
    "4", "305.24"
    "5", "324.17"
    "6", "346.76"
    "7", "376.68"
    "8", "416."
    "9", "465.75"
    "10", "526.20"



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

Analytical solution.


Bibliographical references
---------------------------


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


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