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The Scientific World Journal
Volume 2014 (2014), Article ID 194529, 13 pages
Research Article

Free Vibration Analysis of DWCNTs Using CDM and Rayleigh-Schmidt Based on Nonlocal Euler-Bernoulli Beam Theory

1School of Engineering, Viale dell’Ateneo Lucano 10, 85100 Potenza, Italy
2Department of Structures for Engineering and Architecture, Via Forno Vecchio 36, 80134 Naples, Italy

Received 14 August 2013; Accepted 20 October 2013; Published 25 February 2014

Academic Editors: G. J. Gibbons, P. Lonetti, G. Nikas, and B. F. Yousif

Copyright © 2014 Maria Anna De Rosa and Maria Lippiello. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The free vibration response of double-walled carbon nanotubes (DWCNTs) is investigated. The DWCNTs are modelled as two beams, interacting between them through the van der Waals forces, and the nonlocal Euler-Bernoulli beam theory is used. The governing equations of motion are derived using a variational approach and the free frequencies of vibrations are obtained employing two different approaches. In the first method, the two double-walled carbon nanotubes are discretized by means of the so-called “cell discretization method” (CDM) in which each nanotube is reduced to a set of rigid bars linked together by elastic cells. The resulting discrete system takes into account nonlocal effects, constraint elasticities, and the van der Waals forces. The second proposed approach, belonging to the semianalytical methods, is an optimized version of the classical Rayleigh quotient, as proposed originally by Schmidt. The resulting conditions are solved numerically. Numerical examples end the paper, in which the two approaches give lower-upper bounds to the true values, and some comparisons with existing results are offered. Comparisons of the present numerical results with those from the open literature show an excellent agreement.