Natural frequencies
by Dr. Jiří Plešek
Problem description
Calculate the lowest ten natural frequencies of the cantilever beam shown. Compare the FEM results with the theoretical beam solution.
Material properties
$E=2\times10^5\text{ MPa},$ $\nu=0.3,$ $\rho=7800\text{ kg/m$^3.$}$
Solution
The results of the computation are shown in the table below.
| # | $f\text{ [Hz]}$ | mode | ||||
|---|---|---|---|---|---|---|
| theory | BEAM56 | BEAM61 | BEAM53 | BEAM71 | ||
| 1. | $8.18$ | $8.55$ | $8.18$ | $8.18$ | $8.45$ | bending |
| 2. | $16.36$ | $16.93$ | $16.36$ | $16.36$ | $16.85$ | bending |
| 3. | $51.26$ | $58.04$ | $51.32$ | $51.32$ | $55.00$ | bending |
| 4. | $102.53$ | $113.84$ | $112.65$ | $102.65$ | $112.03$ | bending |
| 5. | $143.54$ | $190.36$ | $147.91$ | $144.65$ | $174.56$ | bending |
| 6. | $365.61$ | $312.76$ | $285.36$ | $354.23$ | ||
| 7. | $287.08$ | $491.76$ | $360.57$ | $289.30$ | $422.37$ | bending |
| 8. | $636.44$ | $552.92$ | $530.75$ | $662.06$ | ||
| 9. | $940.21$ | $706.11$ | $570.72$ | $826.68$ | ||
| 10. | $702.21$ | $1277.15$ | $928.46$ | $706.73$ | $914.59$ | torsion |