DLS 04: Circular plate fixed at inner edge – eigenfrequencies

Description

Material		Modulus of elasticity	$E=200 000 N/mm^2$
		Density	$\rho=7800 kg/m^3$
		Poisson’s ratio	$\nu= 0.3$
Geometry	Cross-section	Thickness	$e = 10mm$
Geometry	Boundary conditions	Along the inner circle	Fixed support
Mesh		Maximum element size	$0.20m$
Mesh		Minimum element size	$0.00m$

Results

Handcalculation

$f_i=\frac{1}{2\cdot \pi \cdot R^2}\cdot \lambda_{ij} \cdot \sqrt{\frac{E \cdot e^2}{12\cdot (1-\nu^2) }}$

$f_i=\frac{1}{2\cdot \pi \cdot R^2}\cdot \begin{bmatrix}13.0 \\ 13.3 \\ 14.7 \\ 18.5 \\ 85.1 \\ 86.7 \\ 91.7 \\ 100.0 \end{bmatrix}\cdot \frac{h}{R^2}\cdot \lambda_{ij} \cdot \sqrt{\frac{E \cdot e^2}{12\cdot (1-\nu^2) }}= \begin{bmatrix}7.93 \\ 8.11 \\ 8.96 \\ 11.28 \\ 51.89 \\ 52.86 \\ 55.91 \\ 60.97 \end{bmatrix} Hz$

Eigen mode	Independent reference	Diamonds	Difference
1	7.93Hz	8.00Hz (M1)	0.93%
2	8.11Hz	8.15Hz (M2)	0.51%
3	8.96Hz	9.01Hz (M4)	0.53%
4	11.28Hz	11.38Hz (M6)	0.89%
5	51.89Hz	53.29Hz (M19)	2.71%
6	52.86Hz	53.42Hz (M20)	1.06%
7	55.91Hz	54.23Hz (M22)	-3.00%
8	60.97Hz	57.57Hz (M24)	-5.58%

The difference between the first 4 and last 4 eigen frequencies are the type of shape mode. There are a few shape modes (accompanied by an eigen frequency) between 11.28 en 51.89Hz which are not listed in the table above.

References

Mécaniciens, S. F. D. (1990). Guide de validation des progiciels de calcul des structures: SDLS 04: plaque mince annulaire encastrée sur un moyeu
The dimensions of the Afnor example were too small for Diamonds. So the radius $R$ and young’s modulus $E$ have been adjusted, and the results are recalculated accordingly.