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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 106
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Paper 32

Eigenfrequencies and Vibration Modes of Carbon Nanotubes

M. Strozzi1, L.I. Manevitch2, V.V. Smirnov2, D.S. Shepelev2 and F. Pellicano1

1Department of Engineering "Enzo Ferrari", University of Modena and Reggio Emilia, Modena, Italy
2N.N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow, Russia

Full Bibliographic Reference for this paper
M. Strozzi, L.I. Manevitch, V.V. Smirnov, D.S. Shepelev, F. Pellicano, "Eigenfrequencies and Vibration Modes of Carbon Nanotubes", in , (Editors), "Proceedings of the Twelfth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 32, 2014. doi:10.4203/ccp.106.32
Keywords: nanotubes, vibration, eigenvalues..

The work, described in this paper, is concerned with the analysis of low-frequency linear vibrations of single-walled nanotubes (SWNTs): two approaches are presented: a fully analytical method based on a simplified theory and a semianalytical method based on the theory of thin shells. The semi-analytical approach (referred to as the "numerical approach") is based on the Sanders-Koiter shell theory and the Rayleigh-Ritz numerical procedure. The nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields, which are expanded by means of a double mixed series based on Chebyshev polynomials. The Rayleigh-Ritz method is then applied to obtain numerically approximate natural frequencies and mode shapes. The second approach is based on a reduced version of the Sanders-Koiter shell theory, obtained by assuming small ring and tangential shear deformations. These assumptions enable both the longitudinal and the circumferential displacement fields to be condensed. A fourth-order partial differential equation for the radial displacement field is derived. Eigenfunctions are formally obtained analytically, then the numerical solution of the dispersion equation gives the natural frequencies and the corresponding normal modes. The methods are fully validated by comparing the natural frequencies of the SWNTs with data available in the literature, namely: experiments, molecular dynamics simulations and finite element analyses. A comparison between the results of the numerical and analytical approach is carried out in order to check the accuracy of the results. It is worthwhile to stress that the analytical model enables the results to be obtained with very little computational effort. On the other hand the numerical approach is able to handle the most realistic boundary conditions of the SWNTs (free-free and clamped-free) with high accuracy. Both methods are suitable for a forthcoming extension to multi-walled nanotubes and nonlinear vibrations.

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