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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
Boundary conditions Along the inner circle Fixed support
Mesh Maximum element size 0.20m
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 Shape 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.
  • Tested in Diamonds 2024r01.

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