Effects of Elastic Restraints on the Fundamental Frequency of Nonlocal Nanobeams with Tip Mass

2019 ◽  
Vol 24 (3) ◽  
pp. 520-530
Author(s):  
Malesela K. Moutlana ◽  
Sarp Adali
Keyword(s):  
Fundamental Frequency ◽  
Scale Effects ◽  
Beam Theory ◽  
Nonlocal Theory ◽  
Small Scale ◽  
Nano Scale ◽  
Scale Structures ◽  
Elastically Restrained ◽  
Elastic Restraints ◽  
Tip Mass

The fundamental frequencies of an elastically restrained nanobeam with a tip mass are studied based on the nonlocal Euler-Bernoulli beam theory. The nanobeam has a torsional spring at one end and a translational spring at the other end where a tip mass is attached. The aim is to model a tapping mode atomic force microscope (TM-AFM), which can be utilized in imaging and the manufacture of Nano-scale structures. A TM-AFM uses high frequency oscillations to remove material, shape structures or scan the topology of a Nano-scale structure. The nonlocal theory is effective at modelling Nano-scale structures, as it takes small scale effects into account. Torsional elastic restraints can model clamped and pinned boundary conditions, as their stiffness values change between zero and infinity. The effects of the stiffness of the elastic restraints, tip mass and the small-scale parameter on the fundamental frequency are investigated numerically.

AN INVESTIGATION ON THE CHARACTERISTICS OF BENDING, BUCKLING AND VIBRATION OF NANOBEAMS VIA NONLOCAL BEAM THEORY

2014 ◽  
Vol 11 (06) ◽  
pp. 1350085 ◽  
Author(s):  
SOUMIA BENGUEDIAB ◽  
ABDELWAHED SEMMAH ◽  
FOUZIA LARBI CHAHT ◽  
SOUMIA MOUAZ ◽  
ABDELOUAHED TOUNSI
Keyword(s):  
Scale Effects ◽  
Constitutive Relations ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Small Scale ◽  
Shear Stresses ◽  
Nonlocal Timoshenko Beam Theory ◽  
Walled Carbon Nanotubes

In the present study, a nonlocal hyperbolic shear deformation theory is developed for the static flexure, buckling and free vibration analysis of nanobeams using the nonlocal differential constitutive relations of Eringen. The theory, which does not require shear correction factor, accounts for both small scale effects and hyperbolic variation of shear strains and consequently shear stresses through the thickness of the nanobeam. The equations of motion are derived from Hamilton's principle. Analytical solutions for the deflection, buckling load and natural frequency are presented for a simply supported nanobeam, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory and Reddy beam theories. Present solutions can be used for the static and dynamic analyses of single-walled carbon nanotubes.

scholarly journals Nonlocal Torsional Vibration of Elliptical Nanorods with Different Boundary Conditions

Vibration ◽  
2020 ◽  
Vol 3 (3) ◽  
pp. 189-203 ◽  
Author(s):  
Farshad Khosravi ◽  
Seyyed Amirhosein Hosseini ◽  
Babak Alizadeh Hamidi ◽  
Rossana Dimitri ◽  
Francesco Tornabene
Keyword(s):  
Boundary Conditions ◽  
Torsional Vibration ◽  
Scale Effects ◽  
Nonlocal Theory ◽  
Equation Of Motion ◽  
Small Scale ◽  
Different Boundary Conditions ◽  
Design And Optimization ◽  
Preliminary Validation ◽  
Aerospace Systems

This work aims at investigating the free torsional vibration of one-directional nanostructures with an elliptical shape, under different boundary conditions. The equation of motion is derived from Hamilton’s principle, where Eringen’s nonlocal theory is applied to analyze the small-scale effects. The analytical Galerkin method is employed to rewrite the equation of motion as an ordinary differential equation (ODE). After a preliminary validation check of the proposed formulation, a systematic study investigates the influence of the nonlocal parameters, boundary conditions, geometrical and mechanical parameters on the natural frequency of nanorods; the objective is to provide useful findings for design and optimization purposes of many nanotechnology applications, such as, nanodevices, actuators, sensors, rods, nanocables, and nanostructured aerospace systems.

scholarly journals Free Vibration of Piezo-Nanowires Using Timoshenko Beam Theory with Consideration of Surface and Small Scale Effects

2014 ◽  
Vol 61 (1) ◽  
pp. 139-152 ◽  
Author(s):  
Atta Oveisi
Keyword(s):  
Timoshenko Beam ◽  
Surface Effect ◽  
Scale Effects ◽  
Closed Form Solution ◽  
Beam Theory ◽  
Local Effect ◽  
Form Solution ◽  
Timoshenko Beam Theory ◽  
Small Scale ◽  
Nonlocal Timoshenko Beam Theory

Abstract This paper investigates the influence of surface effects on free transverse vibration of piezoelectric nanowires (NWs). The dynamic model of the NW is tackled using nonlocal Timoshenko beam theory. By implementing this theory with consideration of both non-local effect and surface effect under simply support boundary condition, the natural frequencies of the NW are calculated. Also, a closed form solution is obtained in order to calculate fundamental buckling voltage. Finally, the effect of small scale effect on residual surface tension and critical electric potential is explored. The results can help to design piezo-NW based instruments.

Fundamental frequencies of a nano beam used for atomic force microscopy (AFM) in tapping mode

MRS Advances ◽  
2018 ◽  
Vol 3 (42-43) ◽  
pp. 2617-2626 ◽  
Author(s):  
MALESELA K. MOUTLANA ◽  
SARP ADALI
Keyword(s):  
Atomic Force Microscopy ◽  
Degree Of Freedom ◽  
Small Scale ◽  
Single Degree Of Freedom ◽  
Tapping Mode ◽  
Force Microscopy ◽  
Single Degree ◽  
Nano Scale ◽  
Atomic Force ◽  
Tip Mass

ABSTRACTIn this study we investigate the motion of a torsionally restrained beam used in tapping mode atomic force microscopy (TM-AFM), with the aim of manufacturing at nano-scale. TM-AFM oscillates at high frequency in order to remove material or shape nano structures. Euler-Bernoulli theory and Eringen’s theory of non-local continuum are used to model the nano machining structure composed of two single degree of freedom systems. Eringen’s theory is effective at nano-scale and takes into account small-scale effects. This theory has been shown to yield reliable results when compared to modelling using molecular dynamics.The system is modelled as a beam with a torsional boundary condition at one end; and at the free end is a transverse linear spring attached to the tip. The other end of the spring is attached to a mass, resulting in a single degree of freedom spring-mass system. The motion of the tip of the beam and tip mass can be investigated to observe the tip frequency response, displacement and contact force. The beam and spring–mass frequencies contain information about the maximum displacement amplitude and therefore the sample penetration depth and this allows

scholarly journals Dynamic Behavior of Multi-Layered Viscoelastic Nanobeam System Embedded in a Viscoelastic Medium with a Moving Nanoparticle

2016 ◽  
Vol 33 (5) ◽  
pp. 559-575 ◽  
Author(s):  
Sh. Hosseini Hashemi ◽  
H. Bakhshi Khaniki
Keyword(s):  
Dynamic Behavior ◽  
Viscoelastic Medium ◽  
Scale Effects ◽  
Nonlocal Parameter ◽  
Nonlocal Theory ◽  
Function Technique ◽  
Small Scale ◽  
Transform Method ◽  
Damping Parameter ◽  
Moving Nanoparticle

AbstractIn this paper, dynamic behavior of multi-layered viscoelastic nanobeams resting on a viscoelastic medium with a moving nanoparticle is studied. Eringens nonlocal theory is used to model the small scale effects. Layers are coupled by Kelvin-Voigt viscoelastic medium model. Hamilton's principle, eigen-function technique and the Laplace transform method are employed to solve the governing differential equations. Analytical solutions for transverse displacements of double-layered is presented for both viscoelastic nanobeams embedded in a viscoelastic medium and without it while numerical solution is achieved for higher layered nanobeams. The influences of the nonlocal parameter, stiffness and damping parameter of medium, internal damping parameter and number of layers are studied while the nanoparticle passes through. Presented results can be useful in analysing and designing nanocars, nanotruck moving on surfaces, racing nanocars etc.

scholarly journals Nonlocal Elasticity Theory for Transient Analysis of Higher-Order Shear Deformable Nanoscale Plates

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Woo-Young Jung ◽  
Sung-Cheon Han
Keyword(s):  
Dynamic Response ◽  
Shear Deformation ◽  
Nonlocal Elasticity ◽  
Deformation Theory ◽  
Transient Analysis ◽  
Shear Deformation Theory ◽  
Nonlocal Theory ◽  
Higher Order ◽  
Small Scale ◽  
Nano Scale

The small scale effect on the transient analysis of nanoscale plates is studied. The elastic theory of the nano-scale plate is reformulated using Eringen’s nonlocal differential constitutive relations and higher-order shear deformation theory (HSDT). The equations of motion of the nonlocal theories are derived for the nano-scale plates. The Eringen’s nonlocal elasticity of Eringen has ability to capture the small scale effects and the higher-order shear deformation theory has ability to capture the quadratic variation of shear strain and consequently shear stress through the plate thickness. The solutions of transient dynamic analysis of nano-scale plate are presented using these theories to illustrate the effect of nonlocal theory on dynamic response of the nano-scale plates. On the basis of those numerical results, the relations between nonlocal and local theory are investigated and discussed, as are the nonlocal parameter, aspect ratio, side-to-thickness ratio, nano-scale plate size, and time step effects on the dynamic response. In order to validate the present solutions, the reference solutions are employed and examined. The results of nano-scale plates using the nonlocal theory can be used as a benchmark test for the transient analysis.

scholarly journals Buckling of elastically restrained nonlocal carbon nanotubes under concentrated and uniformly distributed axial loads

2019 ◽  
Vol 10 (1) ◽  
pp. 145-152
Author(s):  
Mouafo Teifouet Armand Robinson ◽  
Sarp Adali
Keyword(s):  
Carbon Nanotubes ◽  
Numerical Solutions ◽  
Ritz Method ◽  
Small Scale ◽  
Axial Loads ◽  
Weak Formulation ◽  
Compressive Loads ◽  
Method Of Solution ◽  
Elastically Restrained ◽  
Elastic Restraints

Abstract. Buckling of elastically restrained carbon nanotubes is studied subject to a combination of uniformly distributed and concentrated compressive loads. Governing equations are based on the nonlocal model of carbon nanotubes. Weak formulation of the problem is formulated and the Rayleigh quotients are obtained for distributed and concentrated axial loads. Numerical solutions are obtained by Rayleigh–Ritz method using orthogonal Chebyshev polynomials. The method of solution is verified by checking against results available in the literature. The effect of the elastic restraints on the buckling load is studied by counter plots in term of small-scale parameter and the spring constants.

An analytical approach for vibration analysis of laminated orthotropic beam based on nonlocal theory

2018 ◽  
Vol 233 (10) ◽  
pp. 3633-3648
Author(s):  
Saeed Khadem Moshir ◽  
Hamidreza Eipakchi
Keyword(s):  
Numerical Solutions ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Nonlocal Theory ◽  
Free Vibrations ◽  
Small Scale ◽  
Laminated Beam ◽  
Analytical Perturbation

On the basis of the first-order shear deformation beam theory, free vibrations and dynamic response of orthotropic laminated beam subjected to transient and harmonic loading have been studied based on Eringen’s nonlocal elasticity theory. Three coupled nonlinear governing partial differential equations of motion are derived using Hamilton’s principle. The purely analytical perturbation method as well as the method of multiple scales are used for the solution. A parametric study is carried out to realize the effect of small-scale and axial static load on the natural frequencies, transient, and harmonic responses. In addition, the obtained results have been compared with numerical solutions and literature.

Buckling Analysis of Micro- and Nano-Rods/Tubes Based on Nonlocal Timoshenko Beam Theory Using Chebyshev Polynomials

2010 ◽  
Vol 123-125 ◽  
pp. 619-622 ◽  
Author(s):  
Seyyed Amir Mahdi Ghannadpour ◽  
Bijan Mohammadi
Keyword(s):  
Scale Effects ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Buckling Loads ◽  
Various Boundary Conditions ◽  
Total Potential ◽  
Nonlocal Timoshenko Beam Theory ◽  
Nano Rods ◽  
Comparison Of The Results

This paper presents the elastic buckling behavior of nonlocal micro- and nano- Timoshenko rods/tubes based on Eringen’s nonlocal elasticity theory. The critical buckling loads are obtained using the theorem of minimum total potential energy and Chebyshev polynomial functions. The present method, which uses Rayleigh–Ritz technique in this paper, provides an efficient and extremely accurate buckling solution. Numerical results for a variety of some micro- and nano-rods/tubes with various boundary conditions are given and compared with the available results wherever possible. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is promoted. The small scale effects on the buckling loads of rods/tubes are determined and discussed.

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