NDLI: Free vibration analysis of embedded SWCNTs using DQM based on nonlocal Euler-Bernoulli beam theory

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Free vibration analysis of embedded SWCNTs using DQM based on nonlocal Euler-Bernoulli beam theory

Content Provider	Semantic Scholar
Author	Rosa, María Anna De Lippiello, Maria Martin, H. D. Vairo, Francesco
Copyright Year	2013
Abstract	The present paper deals with the free vibration analysis of single-walled carbon nanotube in an elastic medium based on nonlocal Euler-Bernoulli beam theory. The differential quadrature method (DQM) is applied to finding free frequencies of nano-beam systems having to the ends translational and rotational elastic constraints and taking into account the influence of the rotary inertia. Numerical examples are performed to show the accuracy of the proposed method. Index Terms carbon nanotube, free vibration, nonlocal effect, DQM. I . Introduction Carbon nanotubes (CNTs) constitute a prominent example of nanomaterials and nanostructures, which have attracted attention by many researches due to their extraordinary electrical, physical and mechanical properties. Extensive studies have been conducted on studying the mechanical properties of CNTs such as bending, buckling and post buckling [1-2] and free vibration [3]. Recently, many nonlocal elastic continuum models have been used for studying the mechanical behaviour of CNTs including beam models. Their application to the analysis of CNTs allows to evaluate of the small-scale effects influence, [4-5]. The present paper studies free vibration of embedded SWCNT based on nonlocal Euler-Bernoulli beam theory. The surrounding elastic medium is described as the Winkler and Pasternak models, defined by the kw and kp costants. The Hamilton’s principle is applied to derive the governing equations and boundary conditions, which are solved by using the well-known differential quadrature method (DQM), [6-13]. The influences of the elastic medium coefficients, nonlocal parameter and end supports on the free vibrations characteristics of the SWCNT are described. II . Problem formulation Let us consider a single-walled carbon nanotube with length L, Young modulus E, second moment of area I, mass density  and cross-sectional area A. The nanotube analyzed is assumed to be embedded in an elastic medium and it has to the ends translational kTL, kTR and rotational kRL, kRR elastic constraints. The elastic media is simulated using the Winkler and Pasternak foundation models. According to the nonlocal Euler-Bernoulli beam theory and taking into account of the nonlocal effects and the influence of rotary inertia, the equation of motion, for an embedded SWCNTs, is:                   2 '''' 2 '''' 2 '' p 2 '' w p w EI+ k v , t Ι v , t A+ Ι v , t k k v , t Αv , t k v , 0 z z z z z z t                (1) where v(z,t) is the transverse displacement, kw is a Winklertype foundation parameter, kp is a shear coefficient or a Pasternaktype parameter and η = e0a is the small scale effect with e0a constant appropriate to each material and a an is internal characteristic length. The transverse displacement of the nanotubes can be assumed in the following generalized form:     i t v , v e z t z   (2) and substituting equation (2) into equation (1) yields:                 2 2 2 '''' 2 2 '' p 2 '' 2 w p w EI+ k Ι v A+ Ι v k k v k Α v 0 z z z z                   (3) The boundary conditions are:               2 2 2 ''' 2 ' p w p 2 2 ' TL EI+ k Ι v k k v A Ι v k v 0, 0 z z z z z                   (4)         2 2 2 '' 2 2 p ' RL EI+ k Ι v Αv k v 0, 0 z z z z             (5)               2 2 2 ''' 2 ' p w p 2 2 ' TR EI+ k Ι v k k v A Ι v k v 0, L z z z z z                  (6)         2 2 2 '' 2 2 p ' RR EI+ k Ι v Αv k v 0, L z z z z              (7) It is convenient to map the physical domain [0, L] on the natural Gaussian domain [-1, 1], by means of the following transformation:   2 1 L z z    (8) International Conference on Advanced Computer Science and Electronics Information (ICACSEI 2013) © 2013. The authors Published by Atlantis Press 210 where z is the Cartesian co-ordinate and  its natural counterpart. It follows that the differential equation (3) becomes:                 2 2 2 '''' 2 2 '' p 4 2 2 '' 2 w p w 2 16 4 EI+ k Ι v A+ Ι v L L 4 k k v k Α v 0 L                 
File Format	PDF HTM / HTML
DOI	10.2991/icacsei.2013.52
Alternate Webpage(s)	https://download.atlantis-press.com/article/7532.pdf
Alternate Webpage(s)	https://doi.org/10.2991/icacsei.2013.52
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article

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