Size-dependent free vibration of axially functionally graded tapered nanorods having nonlinear spring constraint with a tip nanoparticle

2019 ◽  
Vol 25 (21-22) ◽  
pp. 2769-2783 ◽  
Author(s):  
Arian Bahrami ◽  
Ali Zargaripoor ◽  
Hamidreza Shiri ◽  
Nima Khosravi
Keyword(s):  
Literature Review ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Nonlocal Theory ◽  
Functionally Graded ◽  
The Other ◽  
Constant Mass ◽  
Axial Vibration ◽  
Nonlinear Spring ◽  
Size Dependent

According to the present literature review, the axial vibration of the axially functionally graded (FG) tapered nanorod with attached nonlinear spring has not been addressed so far. In this study, the axial vibration of the FG tapered nanorod is studied based on Eringen’s nonlocal theory, in which one end of the nanorod is clamped and the other end is attached to a nonlinear spring and a nanoparticle. The influence of different parameters such as the nonlinear spring constant, mass of the nanoparticle, and the nonlocal parameter on the natural frequencies is presented in detail. The solutions via Tables can be utilized as a reliable reference for evaluating the validity of future researches. The presented nonlocal solutions can be helpful for those who are interested in designing micro/nano electromechanical systems.

Free and forced axial vibration of single walled carbon nanotube under linear and harmonic concentrated forces based on nonlocal theory

2020 ◽  
Vol 34 (08) ◽  
pp. 2050067 ◽  
Author(s):  
Farshad Khosravi ◽  
Seyyed Amirhosein Hosseini ◽  
Hasti Hayati
Keyword(s):  
Carbon Nanotube ◽  
Natural Frequencies ◽  
Constitutive Relations ◽  
Nonlocal Parameter ◽  
Nonlocal Theory ◽  
Axial Vibration ◽  
Single Walled Carbon Nanotube ◽  
Single Walled Carbon ◽  
Concentrated Forces

The aim of this paper is to investigate the free and forced axial vibrations under the two various linear and harmonic axial concentrated forces in zigzag single-walled carbon nanotube (SWCNT). Two different boundary conditions, namely clamped–clamped and clamped-free, are established. Eringen’s nonlocal elasticity is employed to justify the nonlocal behavior of constitutive relations. The governing equation and the associated boundary condition are derived based on Hamilton’s principle. In order to solve the derived equation numerically, the assumed modes method is utilized. In the free axial vibration section, the first three natural frequencies are obtained for the various values of the nonlocal parameter. The results are in good agreement in comparison with another study. The fundamental natural frequencies with respect to the nonlocal parameter of the case study as a semiconducting nanotube with boron nitride nanotube (BNNT) as a semiconducting nanotube and SWCNT (5,5) as a metallic nanotube are compared. The effects of the nonlocal parameter, thickness and ratio of the excitation-to-natural frequencies overtime on dimensional and nondimensional axial displacements are studied.

scholarly journals Modeling the Size Effect on the Mechanical Behavior of Functionally Graded Curved Micro/Nanobeam

2018 ◽  
Vol 1 (2) ◽  
Author(s):  
Seyyed Amirhosein Hosseini1 ◽  
O. Rahmani2
Keyword(s):  
Size Effect ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Power Index ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Opening Angle ◽  
Simply Supported ◽  
Fg Nanobeam ◽  
Good Agreement

The size effect on the free vibration and bending of a curved FG micro/nanobeam is studied in this paper. Using the Hamilton principle the differential equations and boundary conditions is derived for a nonlocal Euler-Bernoulli curved micro/nanobeam.  The material properties vary through radius direction. Using the Navier approach an analytical solution for simply supported boundary conditions is obtained where the power index law of FGM, the curved micro/nanobeam opening angle, the effect of aspect ratio and nonlocal parameter on natural frequencies and the radial and tangential displacements were analyzed. It is concluded that increasing the curved micro/nanobeam opening angle results in decreasing and increasing the frequencies and displacements, respectively. To validate the natural frequencies of curved nanobeam, when the radius of it approaches to infinity, is compared with a straight FG nanobeam and showed a good agreement.

Nonlocal axial vibration of the multiple Bishop nanorod system

2018 ◽  
Vol 24 (6) ◽  
pp. 1668-1691 ◽  
Author(s):  
Danilo Z Karličić ◽  
Sadoon Ayed ◽  
Enass Flaieh
Keyword(s):  
Elastic Medium ◽  
Dynamic Models ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Shear Stiffness ◽  
Theoretical Models ◽  
Axial Vibration ◽  
Closed Form Solutions ◽  
Rod Theory

Construction of reliable dynamic models of nanostructures is an important task for design procedures of different nanoresonator devices. Such theoretical models allow as to perform different numerical experiments, which is the key point in the development of advanced nanodevices. This paper presents a new nanoresonator model based on the axial vibration of the elastic multi-nanorod system. It is assumed that the system of multiple nanorods is embedded in an elastic medium. The governing equations of motion of a coupled multi-nanorod system are derived using the Hamilton’s principle, the nonlocal elastic constitutive relation, and Bishop’s rod theory, where effects of inertia of the lateral motion and the shear stiffness are considered. Exact closed form solutions for natural frequencies are obtained for one and multiple nanorod systems with different boundary conditions. Then, results for natural frequencies obtained by the finite difference method are compared with the results obtained analytically. Effects of nonlocal parameter, different rod theories, number of nanorods and stiffness coefficient of an elastic medium on natural frequencies are examined through several numerical examples.

scholarly journals An analytical study of vibration in functionally graded piezoelectric nanoplates: nonlocal strain gradient theory

2019 ◽  
Vol 40 (12) ◽  
pp. 1723-1740 ◽  
Author(s):  
Z. Sharifi ◽  
R. Khordad ◽  
A. Gharaati ◽  
G. Forozani
Keyword(s):  
Strain Gradient ◽  
Analytical Study ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nonlocal Strain Gradient Theory ◽  
Functionally Graded Piezoelectric ◽  
Nonlocal Strain Gradient

AbstractIn this paper, we analytically study vibration of functionally graded piezoelectric (FGP) nanoplates based on the nonlocal strain gradient theory. The top and bottom surfaces of the nanoplate are made of PZT-5H and PZT-4, respectively. We employ Hamilton’s principle and derive the governing differential equations. Then, we use Navier’s solution to obtain the natural frequencies of the FGP nanoplate. In the first step, we compare our results with the obtained results for the piezoelectric nanoplates in the previous studies. In the second step, we neglect the piezoelectric effect and compare our results with those obtained for the functionally graded (FG) nanoplates. Finally, the effects of the FG power index, the nonlocal parameter, the aspect ratio, and the side-tothickness ratio, and the nanoplate shape on natural frequencies are investigated.

Size-dependent free vibration and buckling analysis of sigmoid and power law functionally graded sandwich nanobeams with microstructural defects

2020 ◽  
Vol 234 (18) ◽  
pp. 3667-3688
Author(s):  
Ismail Bensaid ◽  
Ahmed Amine Daikh ◽  
Ahmed Drai
Keyword(s):  
Free Vibration ◽  
Power Law ◽  
Functionally Graded Material ◽  
Volume Fraction ◽  
Nonlocal Parameter ◽  
Shear Deformation Theory ◽  
Nonlocal Elasticity Theory ◽  
Functionally Graded ◽  
Size Dependent ◽  
Graded Material

The investigation conducted in this paper aims to study free vibration and buckling behaviors of size-dependent functionally graded sandwich nanobeams. In order to take into account the small size effects, nonlocal elasticity theory of Eringen's is incorporated. Material properties of the functionally graded sandwich beams are supposed to change continuously through the thickness direction according to two forms of the volume fraction of constituents by power law functionally graded material and sigmoid law functionally graded material. These rules are modified to consider the effect of porosity, which covers four kinds of porosity distributions. Two types of sandwich nanobeams were provided: (a) homogeneous core and functionally graded skins and (b) functionally graded core and homogeneous skins. Third-order shear deformation theory without any shear correction factor in conjunction with Hamilton's principle is used to extract the governing equations of motions of porous functionally graded sandwich nanobeams and then solved analytically for two hinged ends. The effects of nonlocal parameter, length to thickness ratios, material graduation index, amount of porosity, porosity distribution shape, on the nondimensional frequency and critical buckling load of the functionally graded sandwich nanobeams made of porous materials are exhibited by a parametric study.

scholarly journals Free Vibration of Axially Moving Functionally Graded Nanoplates Based on the Nonlocal Strain Gradient Theory

2020 ◽  
Vol 25 (4) ◽  
pp. 587-596
Author(s):  
Cheng Li ◽  
P.Y. Wang ◽  
Q.Y. Luo ◽  
S. Li
Keyword(s):  
Strain Gradient ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Characteristic Scale ◽  
Bending Rigidity ◽  
Material Characteristic ◽  
Nonlocal Strain Gradient

Using the nonlocal strain gradient theory, we explore vibration behaviours of initially bidirectional tensioned functionally graded nanoplates with axial speed. The governing equation of motion can be obtained based on the differential type of nonlocal strain gradient constitutive relation, which characters the dynamics of nanostructures containing kinematic relation. The simply supported boundary constraints on four sides are considered and subsequently the numerical results are determined. It shows that natural frequencies of axially moving nanoplates decrease when increasing the axial speed and the nonlocal parameter. Hence the nonlocal and kinematic factors cause the natural frequencies to decrease or, weaken the equivalent bending rigidity. On the other hand, natural frequencies increase with an increase in the axial tension and material characteristic scale. Hence the strain gradient and tensile stress factors cause the natural frequencies to increase or, strengthen the equivalent bending rigidity. In addition, the natural frequencies get higher with a larger aspect ratio of the functionally graded nanoplate. The larger one between the nonlocal parameter and the material characteristic scale plays a dominant role in the softening and stiffening mechanisms of the nonlocal strain gradient effect. In case of the same magnitude of the nonlocal parameter and the material characteristic scale, the softening and hardening phenomena disappear. The equivalent bending rigidity neither increases nor decreases in such a situation, and its value degenerates to the classical one.

scholarly journals On the Vibrations and Stability of Moving Viscoelastic Axially Functionally Graded Nanobeams

Materials ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 1707 ◽  
Author(s):  
Ali Shariati ◽  
Dong won Jung ◽  
Hamid Mohammad-Sedighi ◽  
Krzysztof Kamil Żur ◽  
Mostafa Habibi ◽  
...  
Keyword(s):  
Critical Velocity ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Dynamical Behavior ◽  
Functionally Graded ◽  
Fine Tuning ◽  
Dynamical Response ◽  
Power Law Distribution ◽  
Axially Moving ◽  
The Stability

In this article, size-dependent vibrations and the stability of moving viscoelastic axially functionally graded (AFG) nanobeams were investigated numerically and analytically, aiming at the stability enhancement of translating nanosystems. Additionally, a parametric investigation is presented to elucidate the influence of various key factors such as axial gradation of the material, viscosity coefficient, and nonlocal parameter on the stability boundaries of the system. Material characteristics of the system vary smoothly along the axial direction based on a power-law distribution function. Laplace transformation in conjunction with the Galerkin discretization scheme was implemented to obtain the natural frequencies, dynamical configuration, divergence, and flutter instability thresholds of the system. Furthermore, the critical velocity of the system was evaluated analytically. Stability maps of the system were examined, and it can be concluded that the nonlocal effect in the system can be significantly dampened by fine-tuning of axial material distribution. It was demonstrated that AFG materials can profoundly enhance the stability and dynamical response of axially moving nanosystems in comparison to homogeneous materials. The results indicate that for low and high values of the nonlocal parameter, the power index plays an opposite role in the dynamical behavior of the system. Meanwhile, it was shown that the qualitative stability of axially moving nanobeams depends on the effect of viscoelastic properties in the system, while axial grading of material has a significant role in determining the critical velocity and natural frequencies of the system.

A New Quasi 3D Nonlocal Plate Theory for Vibration and Buckling of FGM Nanoplates

2017 ◽  
Vol 09 (01) ◽  
pp. 1750008 ◽  
Author(s):  
Mohammed Sobhy ◽  
Ahmed F. Radwan
Keyword(s):  
Thermal Environment ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Nonlocal Theory ◽  
Functionally Graded ◽  
Geometrical Parameters ◽  
Motion Equations ◽  
Nonlocal Equations ◽  
Power Law Index

This paper presents the analyses of free vibration and buckling of functionally graded (FG) nanoplates in thermal environment by using a new quasi-3D nonlocal hyperbolic plate theory in which both shear and normal strains are included. The nonlocal equations of motion for the present problem are derived from Hamilton’s principle. For simply-supported boundary conditions, Navier’s approach is utilized to solve the motion equations. Eringen’s nonlocal theory is employed to capture the effect of the nonlocal parameter on natural frequency and buckling of the FGM nanoplates. Numerical results of the present formulation are compared with those predicted by other theories available in the open literature to explain the accuracy of the suggested theory that contains the shear deformation and thickness stretching. Other numerical examples are also presented to show the influences of the nonlocal coefficient, power law index and geometrical parameters on the vibration and buckling load of FGM nanoplates.

Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory

2018 ◽  
Vol 233 (1) ◽  
pp. 287-301 ◽  
Author(s):  
Behrouz Karami ◽  
Davood Shahsavari ◽  
Li Li ◽  
Moein Karami ◽  
Maziar Janghorban
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Shear Deformation Theory ◽  
Thermal Buckling ◽  
Second Order ◽  
Functionally Graded ◽  
Electric Voltage ◽  
Size Dependent

The effective elastic-piezoelectric properties of nanostructures have been shown to be strongly size-dependent. In this paper, a nonlocal second-order shear deformation formulation is presented to study the size-dependent thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core. Temperature is considered as uniform and nonlinear distributions across plate’s thickness direction. Based on the developed nonlocal second-order shear deformation theory, the size-dependent equations of motion are derived. The nonlocal thermal buckling responses of simply supported nanoplates are solved via Navier method. The reliability of present approach is verified by comparing the existing results provided in the open literature. The influences of nonlocal parameter, gradient index, electric voltage, and Winkler–Pasternak parameters on the thermal buckling characteristics of functionally graded nanoplates are examined.

Study on free vibration behavior of rotating bidirectional functionally graded nano-beams based on Eringen’s nonlocal theory

2020 ◽  
Vol 234 (9) ◽  
pp. 1203-1217 ◽  
Author(s):  
Manash Malik ◽  
Debabrata Das
Keyword(s):  
Free Vibration ◽  
Nonlocal Parameter ◽  
Nonlocal Theory ◽  
Functionally Graded ◽  
Numerical Results ◽  
Mode Switching ◽  
Mode Shapes ◽  
Governing Equations ◽  
Vibration Behavior ◽  
Gradient Parameter

Free vibration behavior of rotating bidirectional functionally graded nano-beams is studied based on Eringen’s nonlocal theory. The beam material consists of ceramic and metal constituents, and the material is graded across the length and thickness directions. The mathematical formulation is framed on Euler–Bernoulli beam theory, and the system of governing equations is derived in variational form using Hamilton’s principle. The governing equations are discretized and transformed to an eigen value problem using Ritz method. The model is formulated to study the flapping and lead-lag motions due to free vibration. The model is verified with the available numerical results. The numerical results are presented in non-dimensional frequency-speed plane to study the influence of normalized nonlocal parameter, length gradient parameter, thickness gradient parameter, root radius parameter, and section aspect ratio. Some normalized mode shapes are presented to illustrate the mode switching phenomenon. The mathematical model of a nonlocal rotating bidirectional functionally graded material nano-beam is presented for the first time through this work and the reported results are new of its kind.

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