scholarly journals A Levy Type Solution for Free Vibration Analysis of a Nano-Plate Considering the Small Scale Effect

10.5772/24828 ◽  
2011 ◽  
Author(s):  
E. Jomehzadeh ◽  
A. R.
Keyword(s):  
Free Vibration ◽  
Scale Effect ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Small Scale ◽  
Type Solution ◽  
Small Scale Effect

A Theoretical Approach for Free Vibration Analysis of the Nano-Plates Considering the Small Scale Effect

2010 ◽  
Author(s):  
Emad Jomehzadeh ◽  
Ali Reza Saidi
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Scale Effect ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Small Scale ◽  
Edge Zone ◽  
Wide Range ◽  
Small Scale Effect

The free vibration analysis of a nano-plate is investigated based on the first order shear deformation theory considering the small scale effect. The governing equations of motion are obtained using Hamilton’s principle by considering the nonlocal constitutive equations of Eringen. These coupled partial differential equations are reformulated into two new equations called the edge-zone and interior equations. Analytical solutions are obtained for a nano-plate with Levy boundary conditions. In order to find the natural frequencies of the nano-plate, the various boundary conditions at one direction of the plate should be imposed. Applying these conditions and setting the determinant of the six order coefficient matrix equal to zero, the natural frequencies of the nano-plate are evaluated. Non-dimensional frequency parameters are presented for over a wide range of nonlocal parameters and different boundary conditions. In addition, the effects of nonlocal parameter on the natural frequency of a nano-plate are discussed in details.

Bending, Buckling and Free Vibration Analysis of Size-Dependent Nanoscale FG Beams Using Refined Models and Eringen’s Nonlocal Theory

2020 ◽  
Vol 12 (01) ◽  
pp. 2050007 ◽  
Author(s):  
Atteshamuddin S. Sayyad ◽  
Yuwaraj M. Ghugal
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Volume Fraction ◽  
Constitutive Relations ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Bottom Surface ◽  
Solution Technique ◽  
Small Scale ◽  
Nonlocal Beam

In this study, a theoretical unification of twenty-one nonlocal beam theories are presented by using a unified nonlocal beam theory. The small-scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The present unified theory satisfies traction free boundary conditions at the top and bottom surface of the nanobeam and hence avoids the need of shearing correction factor. Hamilton’s principle is employed to derive the equations of motion. The present unified nonlocal formulation is applied for the bending, buckling and free vibration analysis of functionally graded (FG) nanobeams. The elastic properties of FG material vary continuously by gradually changing the volume fraction of the constituent materials in the thickness direction. Closed-form analytical solutions are obtained by using Navier’s solution technique. Non-dimensional displacements, stresses, natural frequencies and critical buckling loads for FG nanobeams are presented. The numerical results presented in this study can be served as a benchmark for future research.

Energy meaning of small-scale effect on free vibration of carbon nanotubes

2014 ◽  
Vol 10 (5/6) ◽  
pp. 415
Author(s):  
Li Ming ◽  
Zheng Huiming ◽  
Luo Xia ◽  
Liu Yang
Keyword(s):  
Carbon Nanotubes ◽  
Free Vibration ◽  
Scale Effect ◽  
Small Scale ◽  
Small Scale Effect

SMALL SCALE EFFECT ON THE FREE VIBRATION OF MULTI-WALLED CARBON NANOTUBES

2008 ◽  
Vol 22 (28) ◽  
pp. 2769-2777 ◽  
Author(s):  
Y. YAN ◽  
W. Q. WANG ◽  
L. X. ZHANG
Keyword(s):  
Carbon Nanotubes ◽  
Free Vibration ◽  
Scale Effect ◽  
Large Scale ◽  
Small Scale ◽  
Beam Model ◽  
Aspect Ratios ◽  
Small Scale Effect ◽  
Multi Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

This paper is concerned with the free vibration of multi-walled carbon nanotubes (MWCNTs) with simply supported ends. Based on the non-local elasticity theory, Timoshenko beam model with the small scale effect and the van der Waals (vdW) interaction is derived and then solved analytically. The results reveal that the small scale effect is quite significant for small aspect ratios, large scale parameters and high radial vibration modes, whereas it is insensitive to the number of layers of MWCNTs and is weakly-dependent on the wall thickness of MWCNTs.

scholarly journals Nonlocal Vibration Analysis of a Nonuniform Carbon Nanotube with Elastic Constraints and an Attached Mass

Materials ◽  
2021 ◽  
Vol 14 (13) ◽  
pp. 3445
Author(s):  
Maria Anna De Rosa ◽  
Maria Lippiello ◽  
Enrico Babilio ◽  
Carla Ceraldi
Keyword(s):  
Exact Solution ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Numerical Results ◽  
Small Scale ◽  
Concentrated Mass ◽  
Closed Form Solutions

Here, we consider the free vibration of a tapered beam modeling nonuniform single-walled carbon nanotubes, i.e., nanocones. The beam is clamped at one end and elastically restrained at the other, where a concentrated mass is also located. The equation of motion and relevant boundary conditions are written considering nonlocal effects. To compute the natural frequencies, the differential quadrature method (DQM) is applied. The influence of the small-scale parameter, taper ratio coefficient, and added mass on the first natural frequency is investigated and discussed. Some numerical examples are provided to verify the accuracy and validity of the proposed method, and numerical results are compared to those obtained from exact solution. Since the numerical results are in excellent agreement with the exact solution, we argue that DQM provides a simple and powerful tool that can also be used for the free vibration analysis of carbon nanocones with general boundary conditions for which closed-form solutions are not available in the literature.

A new mixed method for nonlinear fuzzy free vibration analysis of nanobeams on nonlinear elastic foundation

2016 ◽  
Vol 24 (24) ◽  
pp. 5765-5773
Author(s):  
AR Vosoughi ◽  
MR Nikoo
Keyword(s):  
Free Vibration ◽  
Elastic Foundation ◽  
Mixed Method ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Small Scale ◽  
Governing Equations ◽  
Nonlinear Elastic Foundation ◽  
Fuzzy Transformation ◽  
Nonlinear Elastic

A new mixed method for nonlinear fuzzy free vibration analysis of nanobeams on nonlinear elastic foundation is introduced. The governing equations are derived based on the first-order shear deformation theory (FSDT) in conjunction with the von-Kármán’s assumptions and the Eringen’s nonlocal elasticity theory. The differential quadrature method (DQM) is employed to discretize the governing equations and the related boundary conditions. The direct displacement control iterative method is used to solve the discretized system of equations. The fuzzy transformation method (FTM) is coupled with the solution to incorporate effects of different uncertainties such as the small scale effect parameter, nonlinear elastic foundation parameters and vibration amplitude of the nanobeam. Applicability, rapid rate of convergence and high accuracy of the presented method are shown and significant effects of the nonlinearity on the response of nanobeams are investigated via solving some examples.

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