DLX 04: Thin circle – fixed at two points – eigen frequencies

Description

Material		Modulus of elasticity	$E=7 200 000 N/mm^2$
Material		Density	$\rho=2700 kg/m^3$
Geometry	Cross-section		$b = 10 mm$
			$h = 5 mm$
	Boundary conditions	In points A at angle $\alpha=120 ^{\circ}$	Fixed support
		Along the entire edge	$R_z, M_x$ and $M_y$ fixed to impose 2D behaviour
Mesh		No. of divisions	8

Results

Handcalculation

$f_i=\frac{1}{2\cdot \pi}\cdot \lambda_i\cdot \frac{h}{R^2}\cdot \sqrt{\frac{E}{12\cdot \rho }}$

$f_i=\frac{1}{2\cdot \pi}\cdot \begin{bmatrix}1.9832 \\ 4.8497 \\ 9.3204 \\ 11.8490 \\ 14.7614 \\ 21.5545 \\ 23.6157 \end{bmatrix}\cdot \frac{h}{R^2}\cdot \sqrt{\frac{E}{12\cdot \rho }}= \begin{bmatrix}23.5 \\ 57.5 \\ 110.6 \\ 140.6 \\ 175.1 \\ 255.7 \\ 280.1 \end{bmatrix}Hz$

Eigen mode	Independent reference	Diamonds	Difference
1	23.5Hz	23.6Hz	-0.31%
2	57.5Hz	57.8Hz	-0.47%
3	110.6Hz	111.1Hz	-0.48%
4	140.6Hz	141.2Hz	-0.45%
5	175.1Hz	175.9Hz	-0.45%
6	255.7Hz	256.8Hz	-0.43%
7	280.1Hz	281.6Hz	-0.52%

References

Mécaniciens, S. F. D. (1990). Guide de validation des progiciels de calcul des structures: SDLL 12: anneau circulaire mince encastré en deux points
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.

Tested in Diamonds 2023r01.

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