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).

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.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

Solving a linear fractional equation with nonlocal boundary conditions based on multiscale orthonormal bases method in the reproducing kernel space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Jiang ◽  
Zhong Chen ◽  
Ning Hu ◽  
Yali Chen
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Reproducing Kernel ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Orthonormal Bases ◽  
Fractional Differential

AbstractIn recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

Sign in / Sign up

Export Citation Format

Share Document