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


.. _RefNumPara__1525_1601233149:


Benchmark solution
---------------------

According to [:ref:`bib1 <bib1>`], the natural vibration frequencies of a rectangular plate supported on all its edges are given by the analytical formula:

:math:`{f}_{\mathit{ij}}=\frac{{\lambda }_{\mathit{ij}}^{2}}{2\pi {L}^{2}}{\left[\frac{E{h}^{3}}{12{\rho }_{s}(1-{\nu }^{2})}\right]}^{\frac{1}{2}}`

with

:math:`{\lambda }_{\mathit{ij}}^{2}={\pi }^{2}\left[{i}^{2}+{\left(\frac{L}{l}\right)}^{2}{j}^{2}\right]`

where

:math:`i` and :math:`j` are the number of half-waves of the modal deformation along the major and minor axes of the plate. :math:`{\rho }_{s}` is the mass per unit area.

Either

:math:`\begin{array}{c}{f}_{11}=\mathrm{17,13}\mathit{Hz}\\ {f}_{21}=\mathrm{35,63}\mathit{Hz}\\ {f}_{12}=\mathrm{50,01}\mathit{Hz}\\ {f}_{31}=\mathrm{66,46}\mathit{Hz}\\ {f}_{22}=\mathrm{68,51}\mathit{Hz}\end{array}`.



The modal deformations are also calculated analytically: the modal displacement :math:`{z}_{\mathit{ij}}` perpendicular to the plate, for mode :math:`(i,j)`, as a function of the coordinate point :math:`(x,y)`, is given by:

:math:`{z}_{\mathit{ij}}(x,y)=\mathrm{sin}\left(\frac{i\pi x}{L}\right)\mathrm{sin}\left(\frac{j\pi y}{l}\right)`.


Case of calculations by substructure: reference solution for each substructure
------------------------------------------------------------------------

Each substructure is a plate of length :math:`l=\mathrm{1,5}m` and width :math:`\frac{L}{2}=1m`, supported on three sides and embedded on one long side, vibrating in flexure.

We show [:ref:`bib1 <bib1>`] that the natural frequencies are equal to:

:math:`{f}_{\mathit{ij}}=\frac{{\lambda }_{\mathit{ij}}^{2}}{2\pi {l}^{2}}{\left[\frac{E{h}^{3}}{12{\rho }_{s}(1-{\nu }^{2})}\right]}^{\frac{1}{2}}`

with :math:`{\lambda }_{11}^{2}=\mathrm{42,53}`, :math:`{\lambda }_{21}^{2}=\mathrm{69,00}`, :math:`{\lambda }_{31}^{2}=\mathrm{116,30}`, :math:`{\lambda }_{12}^{2}=\mathrm{121,00}`

This gives for the first frequencies:

:math:`\begin{array}{}{f}_{11}=\mathrm{47,26}\mathrm{Hz}\\ {f}_{21}=\mathrm{76,57}\mathrm{Hz}\\ {f}_{31}=\mathrm{129,24}\mathrm{Hz}\\ {f}_{12}=\mathrm{134,47}\mathrm{Hz}\end{array}`


Bibliographical reference
-------------------------


1. BLEVINS R.D: Formulas for natural frequency and mode shape. Ed. Krieger 1984.