scholarly journals Torsional vibration of cracked carbon nanotubes with torsional restraints using Eringen’s nonlocal differential model

2018 ◽  
Vol 38 (1) ◽  
pp. 70-87 ◽  
Author(s):  
Mustafa Ö Yayli ◽  
Suheyla Y Kandemir ◽  
Ali E Çerçevik
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Torsional Vibration ◽  
Nonlocal Boundary ◽  
Coefficient Matrix ◽  
Vibration Frequencies ◽  
Sine Series ◽  
Differential Model ◽  
Fourier Sine ◽  
Fourier Sine Series

Free torsional vibration of cracked carbon nanotubes with elastic torsional boundary conditions is studied. Eringen’s nonlocal elasticity theory is used in the analysis. Two similar rotation functions are represented by two Fourier sine series. A coefficient matrix including torsional springs and crack parameter is derived by using Stokes’ transformation and nonlocal boundary conditions. This useful coefficient matrix can be used to obtain the torsional vibration frequencies of cracked nanotubes with restrained boundary conditions. Free torsional vibration frequencies are calculated by using Fourier sine series and compared with the finite element method and analytical solutions available in the literature. The effects of various parameters such as crack parameter, geometry of nanotubes, and deformable boundary conditions are discussed in detail.

An efficient solution method for the longitudinal vibration of nanorods with arbitrary boundary conditions via a hardening nonlocal approach

2016 ◽  
Vol 24 (11) ◽  
pp. 2230-2246 ◽  
Author(s):  
Mustafa Özgür Yayli
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Longitudinal Vibration ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Dynamical Analysis ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions ◽  
Fourier Sine Series ◽  
Nonlocal Approach

The free longitudinal vibration analysis of nanorods (carbon nanotubes) with arbitrary boundaries is presented via a hardening nonlocal approach. Stokes’ transformation incorporated with Fourier sine series is employed for the simulation of nanorod deformation. The Fourier coefficients for nanorods having ends with axial restraints are obtained by the substitution of a deformation function and its derivatives into the governing equation. The explicit expressions are derived for the angular frequencies by using nonlocal boundary conditions in terms of nondimensional parameters. A detailed parametric investigation has been carried out to study the effects of the nonlocal and spring parameters on the size-dependent vibration characteristics of the nanorods. The main objective of this study is to present a general analytical approach for the dynamical analysis of nanorods (carbon nanotubes) with arbitrary boundary conditions (restrained or rigid).

Torsional Vibration Analysis of Carbon Nanotubes Based on the Strain Gradient Theory and Molecular Dynamic Simulations

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
R. Ansari ◽  
R. Gholami ◽  
S. Ajori
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Molecular Dynamic ◽  
Strain Gradient ◽  
Torsional Vibration ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Length Scale ◽  
Molecular Dynamic Simulations ◽  
Dynamic Simulations

In the current study, the torsional vibration of carbon nanotubes is examined using the strain gradient theory and molecular dynamic simulations. The model developed based on this gradient theory enables us to interpret size effect through introducing material length scale parameters. The model accommodates the modified couple stress and classical models when two or all material length scale parameters are set to zero, respectively. Using Hamilton's principle, the governing equation and higher-order boundary conditions of carbon nanotubes are obtained. The generalized differential quadrature method is utilized to discretize the governing differential equation of the present model along with two boundary conditions. Then, molecular dynamic simulations are performed for a series of carbon nanotubes with different aspect ratios and boundary conditions, the results of which are matched with those of the present strain gradient model to extract the appropriate value of the length scale parameter. It is found that the present model with properly calibrated value of length scale parameter has a good capability to predict the torsional vibration behavior of carbon nanotubes.

Reducing the near boundary errors of nonhomogeneous heat equations by boundary consistent methods

2020 ◽  
Author(s):  
Chein-Shan Liu ◽  
Chih-Wen Chang
Keyword(s):  
Heat Equations ◽  
Series Method ◽  
Sine Series ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Sine ◽  
Fourier Cosine Series ◽  
Fourier Sine Series ◽  
Gibbs Phenomena ◽  
Nonhomogeneous Heat

Abstract In the paper, we point out a drawback of the Fourier sine series method to represent a given odd function, where the boundary Gibbs phenomena would occur when the boundary values of the function are non-zero. We modify the Fourier sine series method by considering the consistent conditions on the boundaries, which can improve the accuracy near the boundaries. The modifications are extended to the Fourier cosine series and the Fourier series. Then, novel boundary consistent methods are developed to solve the 1D and 2D heat equations. Numerical examples confirm the accuracy of the boundary consistent methods, accounting for the non-zeros of the source terms and considering the consistency of heat equations on the boundaries, which can not only overcome the near boundary errors but also improve the accuracy of solution about four orders in the entire domain, upon comparing to the conventional Fourier sine series method and Duhamel’s principle.

Bias Free Adaptive Notch Filter Based on Fourier Sine Series

2014 ◽  
Vol E97.A (2) ◽  
pp. 557-564 ◽  
Author(s):  
Kazuki SHIOGAI ◽  
Naoto SASAOKA ◽  
Masaki KOBAYASHI ◽  
Isao NAKANISHI ◽  
James OKELLO ◽  
...  
Keyword(s):  
Notch Filter ◽  
Sine Series ◽  
Adaptive Notch Filter ◽  
Fourier Sine ◽  
Fourier Sine Series

Frequency Equations for Vibration Analysis of Fuel Rod

2005 ◽  
Author(s):  
Hyeong Koo Kim ◽  
Jae Ik Kim ◽  
Kyu Tae Kim ◽  
Moon Saeng Kim
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Frequency Equation ◽  
Sine Series ◽  
Various Boundary Conditions ◽  
Fuel Rod ◽  
Conventional Analysis ◽  
Fourier Sine Series ◽  
Fuel Design ◽  
Parametric Analyses

In this study, the frequency equations for calculating the natural frequencies of the beams with generally restrained boundary conditions by both translational and rotational springs are derived in matrix form using Fourier sine series. In order to show the validation of the solution, numerical results for two degenerate cases are compared with existing results for natural frequency obtained by the conventional analysis. And as a specific application, the natural frequencies of fuel rod for KSNP (Korean Standard Nuclear Plant) fuel assembly are calculated and compared with the external excitations. As a result, the frequency equation derived by present paper seems to be very useful to evaluate the natural frequencies of the double span beams with various boundary conditions. Especially, when some parametric analyses are needed to modify fuel design, the equation can be applied very easily.

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