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Natural frequencies

Problem description: Calculate the lowest ten natural frequencies of the cantilever beam shown. Compare the FEM results with the theoretical beam solution.

Mesh: Use the following element types:
ITE=56--see appendix B.1, four hexahedra.
ITE=61--see appendix B.2, four semi-loofs.
ITE=53--see appendix B.3, four beam elements.
ITE=71--see appendix B.4, two hexahedra, two semi-loofs.

Material properties: $E=2\times 10^5$ MPa, $\nu=0.3$, $\rho=7800$ kg/m³.

Solution: The results of computation detailed in next sections are shown below.

f [Hz]	theory	ITE=56	ITE=61	ITE=53	ITE=71	mode
1.	8.18	8.55	8.21	8.18	8.54	bending
2.	16.36	16.93	16.75	16.36	16.85	bending
3.	51.26	58.04	51.64	51.32	55.83	bending
4.	102.53	113.84	112.65	102.65	112.03	bending
5.	143.54	190.36	147.96	144.65	175.59	bending
6.		365.61	313.00	285.36	354.23
7.	287.08	491.76	360.57	289.30	423.43	bending
8.		636.44	553.53	530.75	826.21
9.		940.21	789.47	570.72	914.59
10.	702.21	1277.15	928.46	706.73	1273.44	torsion