Exact Solution for Natural Frequencies of Clamped-Clamped Double-Walled Carbon Nanotubes

2013 ◽  
pp. 65-83
Keyword(s):  
Carbon Nanotubes ◽  
Exact Solution ◽  
Natural Frequencies ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

Effective Approximations for Natural Frequencies of Double-Walled Carbon Nanotubes Based on Donnell Shell Theory

2011 ◽  
Vol 2 (2) ◽  
Author(s):  
Demetris Pentaras ◽  
Isaac Elishakoff
Keyword(s):  
Carbon Nanotubes ◽  
Carbon Nanotube ◽  
Natural Frequency ◽  
Characteristic Equation ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Analytical Formulas ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes ◽  
First Time

The vibration behavior of double-walled carbon nanotube (DWCNT) is investigated based on Donnell shell theory with van der Waals interaction taken into consideration. In addition, new results are obtained for the natural frequencies of a DWCNT based on a simplified version of Donnell shell theory by neglecting the tangential inertia terms. The resulting reduced characteristic equation for the natural frequency represents the radial mode of vibration. The factor of neglecting tangential inertia relative to the values of frequencies obtained by full Donnell shell theory is also obtained with attendant interesting results. Further possible simplifications of Donnell shell theory are introduced. For the first time in the literature, the effect of neglect of tangential inertia terms in DWCNTs is investigated. Accurate approximate analytical formulas are uncovered for the fundamental natural frequencies and compared with the exact values.

Vibration of Cantilevered Double-Walled Carbon Nanotubes Predicted by Timoshenko Beam Model and Molecular Dynamics

2015 ◽  
Vol 12 (04) ◽  
pp. 1540017 ◽  
Author(s):  
Rumeng Liu ◽  
Lifeng Wang
Keyword(s):  
Carbon Nanotubes ◽  
Molecular Dynamics ◽  
Natural Frequencies ◽  
Md Simulations ◽  
Beam Model ◽  
Timoshenko Beams ◽  
Fundamental Frequencies ◽  
The Difference ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

Vibration of double-walled carbon nanotubes (DWCNTs) with one end fixed and the other end free is studied by using different beam models of continuum mechanics and the molecular dynamics (MD) simulations. The models of the double-Euler beams (DEB) and the double-Timoshenko beams (DTB) of cantilevered case, with the van der Waals interaction between layers taken into consideration, are applied to predict the natural frequencies of DWCNTs. An analytical solution is first obtained for the DTB model with cantilevered boundary condition. The fundamental frequencies obtained by the DEB model and the DTB model are very close, for the relatively long DWCNTs. The MD simulations show that these two models can predict the natural frequencies well. However, the difference between the DEB model and the DTB model becomes obvious, for the vibration of the relatively short DWCNTs. The DTB model can offer a much better prediction than the DEB model when the DWCNT is very short especially for high-order frequencies.

scholarly journals Buckling of double-walled carbon nanotubes

2012 ◽  
Vol 34 (4) ◽  
pp. 217-224 ◽  
Author(s):  
Isaac Elishakoff ◽  
Kévin Dujat ◽  
Maurice Lemaire
Keyword(s):  
Carbon Nanotubes ◽  
Exact Solution ◽  
Approximate Solution ◽  
Carbon Nanotube ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Simply Supported ◽  
The Finite Difference Method ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

In this note we deal with the approximate solution of the buckling problem of a clamped-free double-walled carbon nanotube. First the finite difference method is utilized to solve this case. Then this approach is verified by solving the buckling problem of a double-walled carbon nanotube that is simply supported at both ends for which the exact solution is available.

scholarly journals Influence of the nonlocal parameter on the transverse vibration of double-walled carbon nanotubes

2015 ◽  
Vol 24 (3-4) ◽  
pp. 79-90
Author(s):  
Fernanda de Borbón ◽  
Daniel Ambrosini
Keyword(s):  
Carbon Nanotubes ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Elastic Stiffness ◽  
Beam Model ◽  
Transverse Vibrations ◽  
Proposed Model ◽  
Cubic Polynomials ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

AbstractA high-order nonlocal continuum beam model is proposed, which can be applied to study the transverse vibrations of double-walled carbon nanotubes (DWCNTs), including those that could have initial deformations due to defects or external actions. A beam element is developed adopting Hermite cubic polynomials as shape functions, and mass and elastic stiffness matrix are presented. The influence of the nonlocal parameter on the vibrational properties of DWCNTs is studied. Using the proposed model, it was found that the nonlocal parameter has a strong influence on the natural frequencies.

A Non-Linear Spring Model for Predicting Modal Behavior of Oscillators Built from Double Walled Carbon Nanotubes

2019 ◽  
Vol 60 ◽  
pp. 21-32
Author(s):  
Yan Wen Lin ◽  
Wu Gui Jiang ◽  
Li Ang Chen ◽  
Wen Guang Liu ◽  
Hang Zou
Keyword(s):  
Carbon Nanotubes ◽  
Natural Frequencies ◽  
Geometric Parameters ◽  
Spring Model ◽  
Nonlinear Spring ◽  
Non Linear ◽  
Linear Spring ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes ◽  
Vdw Interaction

A nonlinear spring model is proposed to investigate the oscillation behavior of oscillators based on double-walled carbon nanotubes (DWCNTs) with open end by using the finite element (FE) method, where non-linear spring elements are used to represent the van der Waals (vdW) interaction between tubes. Compared to the linear spring FE model, the proposed non-linear springs can more accurately describe the interaction between nanotubes because the vdW interaction is a kind of strongly non-linear force. The influence of boundary conditions, geometric parameters of the DWCNTs, and the layer spacing of tubes on the natural frequencies is especially studied. Various oscillation modes and the corresponding natural frequencies are obtained. Compared to the results obtained by using the linear spring model, the natural frequencies of oscillators based on DWCNTs are in qualitatively better agreement with those obtained from the analytical method and the molecular dynamics (MD) method. From the FE results, it also can be seen that, DWCNTs is expected to be a nanoscale oscillatory device, and its oscillation mode and natural frequency can be adjusted by changing the geometric parameters and boundary condition of the tubes. The proposed nonlinear spring model is helpful for the design of the nano-oscillators under various conditions.

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