Eigenmodes of a slotted tube

The resonant behaviour of slotted tubes, clamped at one end but free at the other, has been investigated to determine the influence of slot length upon their characteristic normal modes. FEM simulations have been performed for a series of steel tubes, identical apart from having slots of different lengths, but constant equal widths, running parallel to their symmetry axes. Three dimensional visualizations of the results allowed different modal classes to be identified. The lower frequency region of the spectrum was dominated by longitudinal, breathing and transverse modes. Other classes, such as oval modes, appeared progressively as the frequency increased. The results were checked experimentally using ESPI to view the tubes while exciting them using a magnetic transducer. This provided experimental confirmation of the mode classifications. It was found that the frequencies of the transverse modes and of the breathing modes were lowered as the slot was lengthened. The frequencies of the longitudinal modes were almost unaffected by slots of any size. Where modes occurred in degenerate pairs, always as predicted by group representation theory, the splitting increased with slot length. This was by as much as 100% of the lower frequency in the case of higher order transverse modes. From the FEM simulations a previously unknown type of modal form has been observed. These “modes” appeared in families but only in cases with longer slits. They have non-zero amplitude only around the edges of the slit. This discovery may provide a greater understanding of cylinders with cracks present and their failure under loading.

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