2. Benchmark solution#

2.1. 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 \(a\) side plate, leads to:

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

with, for a Poisson’s ratio \(\nu =0.3\):

1°: Plate embedded on one side

2°: Free plate

\(i\)	\({\lambda }_{i}^{2}\)	\(i\)	\({\lambda }_{i}^{2}\)
1	3.492	1 to 6
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: \(t/a<0.1\)

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

2.2. Benchmark results#

Case 1:6 first clean modes

Case 2:11 first natural modes

2.3. Uncertainty about the solution#

Semi—analytical solution.

2.4. Bibliographical references#

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