Vibrations of an elastic cylindrical shell near the lowest cut-off frequency

Kaplunov, J.; Manevitch, L.I.; Smirnov, V.V.

Abstract

A new asymptotic approximation of the dynamic equations in the two-dimensional classical theory of thin-elastic shells is established for a circular cylindrical shell. It governs long wave vibrations in the vicinity of the lowest cut-off frequency. At a fixed circumferential wavenumber, the latter corresponds to the eigenfrequency of in-plane vibrations of a thin almost inextensible ring. It is stressed that the well-known semi-membrane theory of cylindrical shells is not suitable for tackling a near-cut-off behaviour. The dispersion relation within the framework of the developed formulation coincides with the asymptotic expansion of the dispersion relation originating from full two-dimensional shell equations. Asymptotic analysis also enables refining the geometric hypotheses underlying various ad hoc set-ups, including the assumption on vanishing of shear and circumferential mid-surface deformations used in the semi-membrane theory. The obtained results may be of interest for dynamic modelling of elongated cylindrical thin-walled structures, such as carbon nanotubes.

Citation

Kaplunov, J., Manevitch, L., & Smirnov, V. (2016). Vibrations of an elastic cylindrical shell near the lowest cut-off frequency. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2189), https://doi.org/10.1098/rspa.2015.0753

• Journal Article Type: Article
• Acceptance Date: Apr 5, 2016
• Publication Date: May 4, 2016
• Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
• Print ISSN: 1364-5021
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 472
• Issue: 2189
• DOI: https://doi.org/10.1098/rspa.2015.0753
• Keywords: cut-off, shell, elastic, asymptotic, vibration, nanotube
• Publisher URL: https://doi.org/10.1098/rspa.2015.0753