Reference solution
=====================


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

The reference solution is the one given in sheet SDLS04 /89 of the guide VPCS which presents the analytical solution as follows:

The solution of the frequency determinant established from Bessel functions leads to the formula:

:math:`{f}_{\mathrm{ij}}=\frac{1}{2\pi {R}_{e}^{2}}{\lambda }_{\mathrm{ij}}^{2}\sqrt{\frac{E{t}^{2}}{12\rho (1-{\nu }^{2})}}`

With:


* :math:`i` = number of nodal diameters


* :math:`j` = number of nodal circles

and :math:`{\lambda }_{\mathrm{ij}}^{2}` such as:


.. csv-table::

    "**i**
    **j**", "**0**", "**1**", "**2**", "**3**"
    "0", "13.0", "13.3", "13.3", "14.7", "18.5"
    "1", "85.1", "86.7", "86.7", "91.7", "100."


Flexion mode with 2 nodal diameters and 1 nodal circle: :math:`{f}_{\mathrm{2,1}}=\mathrm{559,09}\mathrm{Hz}`


.. image:: images/10000E1400000E2900000E7888368C4D0662053B.svg
    :width: 182
    :height: 186


.. _RefImage_10000E1400000E2900000E7888368C4D0662053B.svg:


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

8 clean modes.


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

Analytical solution.


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


1. A.W. LEISSA, Vibration of Plates, Document NASA SP160, 1969, pp. 19-30.