American Journal of Mechanical Engineering
ISSN (Print): 2328-4102 ISSN (Online): 2328-4110 Website: Editor-in-chief: Kambiz Ebrahimi, Dr. SRINIVASA VENKATESHAPPA CHIKKOL
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American Journal of Mechanical Engineering. 2014, 2(7), 195-198
DOI: 10.12691/ajme-2-7-5
Open AccessArticle

Vibration of Single-Walled Carbon Nanotubes by Using Nonlocal Theory

Jozef Bocko1, and Pavol Lengvarský1

1Department of Applied Mechanics and Mechanical Engineering, Technical University of Košice, Faculty of Mechanical Engineering, Košice, Slovakia

Pub. Date: October 06, 2014

Cite this paper:
Jozef Bocko and Pavol Lengvarský. Vibration of Single-Walled Carbon Nanotubes by Using Nonlocal Theory. American Journal of Mechanical Engineering. 2014; 2(7):195-198. doi: 10.12691/ajme-2-7-5


The free bending vibration of single-walled carbon nanotubes (SWCNTs) is investigated in the present paper. A continuum approach based on nonlocal theory of beam bending is used for natural frequency computation. Analytical solutions of frequency equations are given for four types boundary conditions: clamped-free (C-F), simply-simply supported (S-S), clamped-simply supported (C-S) and clamped-clamped (C-C). The graphical representations of numerical results are shown for the first case of boundary conditions clamped end - free end.

single-walled carbon nanotube vibration beam nonlocal theory free vibration

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[1]  Eringen, A.C., Nonlocal continuum field theories, Springer-Verlag, 2002.
[2]  Grupta, S.S., Barta, R.C., “Continuum structures equivalent in normal mode vibrations to single-waled carbon nanotubes”, Comp. Mater. Sci., 43, 715-723, 2008.
[3]  Harik, V.M., “Mechanics of carbon nanotubes: applicability of the continuum-beam models”, Comp. Mater. Sci., 24, 328-342, 2002.
[4]  Iijima, S., “Helical microtubules of graphitic carbon”, Nature, 345, 56-58, 1991.
[5]  Li, C.Y., Chou, T., “A structural mechanics approach for the analysis of carbon nanotubes”, Int. J. of Solids and Struct., 40, 2487-2499, 2003.
[6]  Lu, J., Chen, H., Lu, P., Zhang, P., “Research of natural frequency of single-walled carbon nanotube”, Chinese J. of Chem. Phys., 20, 525-530, 2007.
[7]  Lu, P., Lee, H.P., Lu, C., Zhang, P.Q., “Application of nonlocal beam model for carbon nanotubes”, Int. J. of Solids and Struct., 44, 5289-5300, 2007.
[8]  Muc, A., Banas, A., Chwal, M., “Free vibrations of carbon nanotubes with defects”, Mech. and Mechan. Eng., 17, 157-166, 2013.
[9]  Reddy, J.N., Pang, S.D., “Nonlocal continuum theories of beams for the analysis of carbon nanotubes”, J. of Appl. Phys., 103, 023511-1-023511-16, 2008.
[10]  Sakhaee-Pour, A., Ahmadian, M.T., Vafai, A. “Vibrational analysis of single-walled carbon nanotubes using beam element”, Thin-walled struct., 47, 646-652, 2009.
[11]  Swain, A., Roy, T., Nanda, B.K., “Vibration behaviour of single walled carbon nanotube using finite element”, Int. J. Theor. and Appl. Res. in Mech. Eng., 2, 129-133, 2013.
[12]  Thongyothee, C., Chucheepsakul, S., “Finite element modeling of van der Waals interaction for elastic stability of multi-walled carbon nanotubes”, Adv. Mater. Res., 55-57, 525-528, 2008.
[13]  Thongyothee, C., Chucheepsakul, S., Li, T., “Nonlocal elasticity theory for free vibration of single-walled carbon nanotubes”, Adv. Mater. Res., 747, 257-260, 2013.
[14]  Tserpes, K.I., Papanikos, P., “Finite element modeling of single-walled carbon nanotubes”, Comp. Part B: Eng., 36, 468-477, 2005.
[15]  Wang, Q., Liew, K.M., “Application of nonlocal continuum mechanics to static analysis of micro-and nano-structures”, Physics Letters A, 363, 236-242, 2007.