Effects of dimensional parameters and various boundary conditions on axisymmetric vibrations of multi-walled carbon nanotubes using a continuum model

2010 ◽  
Vol 81 (8) ◽  
pp. 1129-1140 ◽  
Author(s):  
S. T. Talebian ◽  
M. Tahani ◽  
M. H. Abolbashari ◽  
S. M. Hosseini
Author(s):  
Sarp Adali

Variational principles are derived for multi-walled carbon nanotubes (CNT) undergoing nonlinear vibrations. Two sources of nonlinearity are considered in the continuum modeling of CNTs with the Euler-Bernoulli beam model describing the dynamics of the CNTs. One source is the geometric nonlinearity which may arise as a result of large deflections. The second source is due to van der Waals forces between the nanotubes which can be modeled as a nonlinear force to improve the accuracy of the physical model. After deriving the applicable variational principle, Hamilton’s principle is given. Natural and geometric boundary conditions are derived using the variational formulation of the problem. Several approximate and computational methods of solution such as Rayleigh-Ritz and finite elements employ the variational formulation of the problem and as such these principles are instrumental in obtaining the solutions of vibration problems under complicated boundary conditions.


2021 ◽  
Author(s):  
Wan Nura’in Nabilah Noranuar ◽  
Ahmad Qushairi Mohamad ◽  
Sharidan Shafie ◽  
Ilyas Khan ◽  
Mohd Rijal Ilias ◽  
...  

The study analyzed the heat transfer of water-based carbon nanotubes in non-coaxial rotation flow affected by magnetohydrodynamics and porosity. Two types of CNTs have been considered; single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). Partial differential equations are used to model the problem subjected to the initial and moving boundary conditions. Employing dimensionless variables transformed the system of equations into ordinary differential equations form. The resulting dimensionless equations are analytically solved for the closed form of temperature and velocity distributions. The obtained solutions are expressed in terms of a complementary function error. The impacts of the embedded parameters are graphically plotted in different graphs and are discussed in detail. The Nusselt number and skin friction are also evaluated. The temperature and velocity profiles have been determined to meet the initial and boundary conditions. An augment in the CNTs’ volume fraction increases both temperature and velocity of the nanofluid as well as enhances the rate of heat transport. SWCNTs provides high values of Nusselt number compared to MWCNTs. For verification, a comparison between the present solutions and a past study is conducted and achieved excellent agreement.


2006 ◽  
Author(s):  
Julio R. Claeyssen ◽  
Rosemaira Dalcin Copetti ◽  
Teresa Tsukazan

In this work we consider segmented Euler-Bernoulli beams that can have an internal damping of the type Kelvin-Voight and external viscous damping at the discontinuities of the sections. In the literature, the study of this kind of beams has been sufficiently studied with proportional damping only, however the effects of non-proportional damping has been little studied in terms of modal analysis. The obtaining of the modes of segmented beams can be accomplished with a the state space methodology or with the classical Euler construction of responses. Here, we follow a newtonian approach with the use of the impulse response of beams subject both types of damping. The use of the dynamical basis, generated by the fundamental solution of a differential equation of fourth order, allows to formulate the eigenvalue problem and the shapes of the modes in a compact manner. For this, we formulate in a block manner the boundary conditions and intermediate conditions at the beam and values of the fundamental matrix at the ends of the beam and in the points intermediate. We have chosen a basis generated by a fundamental response and it derivatives. The elements of this basis has the same shape with a convenient translation for each segment. This choice reduce computations with the number of constants to be determined to find only the ones that correspond to the first segment. The eigenanalysis will allow to study forced responses of multi-span Euler-Bernoulli beams under classical and non-classical boundary conditions as well as multi-walled carbon nanotubes (MWNT) that are modelled as an assemblage of Euler-Bernoulli beams connected throughout their length by springs subject to van der Waals interaction between any two adjacent nanotubes.


Acta Naturae ◽  
2011 ◽  
Vol 3 (1) ◽  
pp. 99-106 ◽  
Author(s):  
E A Smirnova ◽  
A A Gusev ◽  
O N Zaitseva ◽  
E M Lazareva ◽  
G E Onishchenko ◽  
...  

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