Modal analysis, plate

Problem. The symmetric first mode and the antisymmetric second mode of a simply supported square plate are compared.

Reference document. The closed-form frequency relation of a simply supported square plate is used.

The frequencies of a simply supported square plate are computed from this relation. Here $D$ is the flexural rigidity of the plate.

f_{mn} = \dfrac{\pi}{2}\left(\dfrac{m^2}{a^2} + \dfrac{n^2}{b^2}\right)\sqrt{\dfrac{D}{\rho\,t}}

Numerical model. The plate is modelled with sixteen by sixteen quadrilateral shell elements and a consistent mass matrix.

Simply supported square plate

Parameters. a = 6 · t = 0.06 · mesh = 16×16

Closed form. The frequencies of a simply supported square plate

f_{mn} = \dfrac{\pi}{2}\left(\dfrac{m^2}{a^2} + \dfrac{n^2}{b^2}\right)\sqrt{\dfrac{D}{\rho t}}

Quantity	Reference	Computed	Deviation
Mode (1,1) frequency [Hz]	5.34396	5.36075	0.314%
Mode (2,1) frequency [Hz]	13.3599	13.5526	1.44%