Variational Principles for Transversely Vibrating Multiwalled Carbon Nanotubes Based on Nonlocal Euler−Bernoulli Beam Model

Nano Letters ◽  
2010 ◽  
Vol 10 (8) ◽  
pp. 3222-3222
Author(s):  
Sarp Adali
Keyword(s):  
Carbon Nanotubes ◽  
Multiwalled Carbon Nanotubes ◽  
Variational Principles ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Multiwalled Carbon ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model

On the forced vibration of carbon nanotubes via a non-local Euler—Bernoulli beam model

2010 ◽  
Vol 224 (2) ◽  
pp. 497-503 ◽  
Author(s):  
P Karaoglu ◽  
M Aydogdu
Keyword(s):  
Carbon Nanotubes ◽  
Forced Vibration ◽  
Beam Theory ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Resonance Frequencies ◽  
Non Local ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model ◽  
Walled Cnts

This article studies the forced vibration of the carbon nanotubes (CNTs) using the local and the non-local Euler—Bernoulli beam theory. Amplitude ratios for the local and the non-local Euler—Bernoulli beam models are given for single- and double-walled CNTs. It is found that the non-local models give higher amplitudes when compared with the local Euler—Bernoulli beam models. The non-local Euler—Bernoulli beam model predicts lower resonance frequencies.

scholarly journals Numerical Solution of Bending of the Beam with Given Friction

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 898
Author(s):  
Michaela Bobková ◽  
Lukáš Pospíšil
Keyword(s):  
Sliding Friction ◽  
Quadratic Function ◽  
Building Construction ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Sustainable Building ◽  
Construction Design ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model ◽  
Fixed Beam

We are interested in a contact problem for a thin fixed beam with an internal point obstacle with possible rotation and shift depending on a given swivel and sliding friction. This problem belongs to the most basic practical problems in, for instance, the contact mechanics in the sustainable building construction design. The analysis and the practical solution plays a crucial role in the process and cannot be ignored. In this paper, we consider the classical Euler–Bernoulli beam model, which we formulate, analyze, and numerically solve. The objective function of the corresponding optimization problem for finding the coefficients in the finite element basis combines a quadratic function and an additional non-differentiable part with absolute values representing the influence of considered friction. We present two basic algorithms for the solution: the regularized primal solution, where the non-differentiable part is approximated, and the dual formulation. We discuss the disadvantages of the methods on the solution of the academic benchmarks.

scholarly journals Variational Principles for Multiwalled Carbon Nanotubes Undergoing Vibrations Based on Nonlocal Timoshenko Beam Theory

2010 ◽  
Vol 2010 ◽  
pp. 1-7 ◽  
Author(s):  
Ismail Kucuk ◽  
Ibrahim S. Sadek ◽  
Sarp Adali
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Multiwalled Carbon Nanotubes ◽  
Timoshenko Beam ◽  
Variational Principles ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Small Scale ◽  
Multiwalled Carbon ◽  
Nonlocal Timoshenko Beam Theory

Variational principles are derived for multiwalled carbon nanotubes undergoing linear vibrations using the semi-inverse method with the governing equations based on nonlocal Timoshenko beam theory which takes small scale effects and shear deformation into account. Physical models based on the nonlocal theory approximate the nanoscale phenomenon more accurately than the local theories by taking small scale phenomenon into account. Variational formulation is used to derive the natural and geometric boundary conditions which give a set of coupled boundary conditions in the case of free boundaries which become uncoupled in the case of the local theory. Hamilton's principle applicable to this case is also given.

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