2. Benchmark solution#

2.1. Benchmark solution#

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

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

with

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

where

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

Either

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

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

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

2.2. Case of calculations by substructure: reference solution for each substructure#

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

We show [bib1] that the natural frequencies are equal to:

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

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

This gives for the first frequencies:

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

2.3. Bibliographical reference#

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