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


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

The reference solution is the one given in sheet SDLS01 /89 of the guide VPCS which presents the calculation method as follows:

The formulation of M.V. BARTON, for a :math:`a` side plate, leads to:

:math:`{f}_{i}=\frac{1}{2\pi {a}^{2}}{\lambda }_{i}^{2}\sqrt{\frac{{\mathrm{Et}}^{2}}{12\rho (1-{\nu }^{2})}}` :math:`i=\mathrm{1,2},\mathrm{...}`

with, for a Poisson's ratio :math:`\nu =0.3`:


.. csv-table::

    "1°: Plate embedded on one side", "2°: Free plate"



.. csv-table::

    ":math:`i` "," :math:`{\lambda }_{i}^{2}` ", "", ":math:`i` "," :math:`{\lambda }_{i}^{2}`"
    "1", "3.492", "", "1 to 6", "0."
    "2", "8.525", "", "7", "13.49"
    "3", "21.43", "", "8", "19.79"
    "4", "27.33", "", "9", "24.43"
    "5", "31.11", "", "10", "35.02"
    "6", "54.44", "", "11", "35.02"


(6 solid body modes at zero frequency).

This reference solution applies to thin plates such as: :math:`t/a<0.1`

The coefficients :math:`{\lambda }_{i}` are established by limited development on the modal deformations of a network of crossed beams (embedded beam—free and free—free).


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

Case 1:6 first clean modes

Case 2:11 first natural modes


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

Semi—analytical solution.


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


1. M.V. BARTON Vibrations of rectangular and skew cantilever plates. — Journal of Applied Mechanics, vol18, p.129—134 (1951)