Differential quadrature solutions for the nonconservative instability of a class of single-walled carbon nanotubes

2018 ◽  
Vol 35 (1) ◽  
pp. 251-267 ◽  
Author(s):  
Maria Anna De Rosa ◽  
Maria Lippiello ◽  
Stefania Tomasiello
Keyword(s):  
Free Vibration ◽  
Scale Effects ◽  
Nonlocal Parameter ◽  
Added Mass ◽  
Differential Quadrature ◽  
Small Scale ◽  
Numerical Schemes ◽  
Content Type ◽  
Single Walled Carbon ◽  
Integration Schemes

Purpose The purpose of the present paper is to investigate the nonconservative instability of a single-walled carbon nanotube (SWCNT) with an added mass through nonlocal theories. The governing equations are discretized by means of the differential quadrature (DQ) rules, as introduced by Bellman and Casti. DQ rules have been largely used in engineering and applied sciences. Recently, they were applied to enhance some numerical schemes, such as step-by-step integration schemes and Picard-like numerical schemes. Design/methodology/approach In the present paper, the DQ rules are used to investigate the nonconservative instability of a SWCNT through nonlocal theories. Findings To show the sensitivity of the SWCNT to the values of added mass and the influence of nonlocal parameter on the fundamental frequencies values, some numerical examples have been performed and discussed. Yet, the effect of the different boundary conditions on the instability behaviour has been investigated. The validity of the present model has been confirmed by comparing some results against the ones available in literature. Originality/value Applying the nonlocal elasticity theory, this paper presents a re-formulation of Hamilton’s principle for the free vibration analysis of a uniform Euler–Bernoulli nanobeam. The main purpose of this paper is to investigate the free vibration response of an SWCNT with attached mass and for various values of small scale effects.

Analytical Treatment of the Free Vibration of Single-Walled Carbon Nanotubes Based on the Nonlocal Flugge Shell Theory

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
R. Ansari ◽  
H. Rouhi
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Scale Effects ◽  
Elastic Shell ◽  
Nonlocal Parameter ◽  
Single Walled Carbon Nanotubes ◽  
Small Scale ◽  
Resonant Frequencies ◽  
Single Walled Carbon ◽  
Walled Carbon Nanotubes

In the current work, the vibration characteristics of single-walled carbon nanotubes (SWCNTs) under different boundary conditions are investigated. A nonlocal elastic shell model is utilized, which accounts for the small scale effects and encompasses its classical continuum counterpart as a particular case. The variational form of the Flugge type equations is constructed to which the analytical Rayleigh–Ritz method is applied. Comprehensive results are attained for the resonant frequencies of vibrating SWCNTs. The significance of the small size effects on the resonant frequencies of SWCNTs is shown to be dependent on the geometric parameters of nanotubes. The effectiveness of the present analytical solution is assessed by the molecular dynamics simulations as a benchmark of good accuracy. It is found that, in contrast to the chirality, the boundary conditions have a significant effect on the appropriate values of nonlocal parameter.

scholarly journals Double mode model of size-dependent chaotic vibrations of nanoplates based on the nonlocal elasticity theory

2021 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Grzegorz Kudra ◽  
Olga Mazur
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Scale Effects ◽  
Plate Theory ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Phase Portraits ◽  
Small Scale ◽  
Governing System ◽  
Account Due

AbstractIn this paper vibrations of the isotropic micro/nanoplates subjected to transverse and in-plane excitation are investigated. The governing equations of the problem are based on the von Kármán plate theory and Kirchhoff–Love hypothesis. The small-size effect is taken into account due to the nonlocal elasticity theory. The formulation of the problem is mixed and employs the Airy stress function. The two-mode approximation of the deflection and application of the Bubnov–Galerkin method reduces the governing system of equations to the system of ordinary differential equations. Varying the load parameters and the nonlocal parameter, the bifurcation analysis is performed. The bifurcations diagrams, the maximum Lyapunov exponents, phase portraits as well as Poincare maps are constructed based on the numerical simulations. It is shown that for some excitation conditions the chaotic motion may occur in the system. Also, the small-scale effects on the character of vibrating regimes are illustrated and discussed.

A Novel Refined Plate Theory for Free Vibration Analyses of Single-Layered Graphene Sheets Lying on Winkler-Pasternak Elastic Foundations

2019 ◽  
Vol 58 ◽  
pp. 151-164 ◽  
Author(s):  
Fatima Boukhatem ◽  
Aicha Bessaim ◽  
Abdelhakim Kaci ◽  
Abderrahmane Mouffoki ◽  
Mohammed Sid Ahmed Houari ◽  
...  
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Elastic Foundation ◽  
Displacement Field ◽  
Scale Effects ◽  
Plate Theory ◽  
Small Scale ◽  
Graphene Sheets ◽  
Refined Plate Theory ◽  
Foundation Parameters

In this article, the analyses of free vibration of nanoplates, such as single-layered graphene sheets (SLGS), lying on an elastic medium is evaluated and analyzed via a novel refined plate theory mathematical model including small-scale effects. The noteworthy feature of theory is that the displacement field is modelled with only four unknowns, which is even less than the other shear deformation theories. The present one has a new displacement field which introduces undetermined integral variables, the shear stress free condition on the top and bottom surfaces of the plate is respected and consequently, it is unnecessary to use shear correction factors. The theory involves four unknown variables, as against five in case of other higher order theories and first-order shear deformation theory. By using Hamilton’s principle, the nonlocal governing equations are obtained and they are solved via Navier solution method. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, nonlocal parameter, and elastic foundation parameters are all examined. From this work, it can be observed that the small-scale effects and elastic foundation parameters are significant for the natural frequency.

Free vibration and buckling analysis of elastically supported transversely inhomogeneous functionally graded nanoplate in thermal environment using Rayleigh–Ritz method

2020 ◽  
pp. 107754632096693
Author(s):  
Piyush P Singh ◽  
Mohammad S Azam
Keyword(s):  
Free Vibration ◽  
Thermal Environment ◽  
Ritz Method ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Small Scale ◽  
Pasternak Foundation ◽  
Differential Model ◽  
Aspect Ratios ◽  
Elastically Supported

In this study, free vibration and buckling behaviors of a functionally graded nanoplate supported by the Winkler–Pasternak foundation using a nonlocal classical plate theory are investigated. Eringen’s nonlocal differential model has been used for considering the small-scale effect. The properties of the functionally graded nanoplate are considered to vary transversely following the power law. The governing vibration and buckling equations of an elastically supported functionally graded nanoplate have been derived using the principle of virtual work, and the solution is obtained using the Rayleigh–Ritz method and characteristic polynomials. The advantage of this method is that it disposes of all the drawbacks regarding edge constraints. The objective of the article is to see the effect of edge constraints, aspect ratios, material property exponent, nonlocal parameter, and foundation parameters on the nondimensionalized frequency and the buckling load of an embedded functionally graded nanoplate in a thermal environment. The study highlights that the nonlocal effect is pronounced for higher modes and/or higher aspect ratios and need to be considered for the analysis of the nanoplate. Further, it is observed that the effect of the Pasternak foundation is prominent on nondimensionalized frequencies and buckling of the functionally graded nanoplate.

Investigation of viscous fluid flow and dynamic stability of CNTs subjected to axial harmonic load coupled using Bolotin’s method

2019 ◽  
Vol 30 (6) ◽  
pp. 3435-3462 ◽  
Author(s):  
Mohammad Hashemian ◽  
Amir Homayoun Vaez ◽  
Davood Toghraie
Keyword(s):  
Viscous Fluid ◽  
Dynamic Stability ◽  
Knudsen Number ◽  
Static Load ◽  
Scale Effects ◽  
Fluid Velocity ◽  
Small Scale ◽  
Load Factor ◽  
Harmonic Load ◽  
Content Type

Purpose The dynamic stability of nano-tubes is an important issue in engineering applications. Dynamic stability of anti-symmetric coupled-carbon nanotubes (C-CNTs)-systems in thermal environment is presented in this paper. In this system, the top and bottom CNTs are subjected to axial harmonic load and action of the viscous fluid, respectively. Design/methodology/approach The coupling and surrounding mediums of the CNTs are simulated by visco-Pasternak foundation containing the spring, shear and damper coefficients. Based on the Timoshenko beam theory and Hamilton’s principle, the coupled motion equations are derived considering size effects using Eringen’s nonlocal theory. Using the exact solution in conjunction with Bolotin’s method, the dynamic instability region (DIR) of the coupled structure is obtained. The effects of various parameters such as small scale parameter, Knudsen number, fluid velocity, static load factor, temperature change, surrounding medium and nanotubes aspect ratio are shown on the DIR of the coupled system. Findings Results indicate that considering parameters such as small scale effects, static load factor, Knudsen number and fluid velocity shifts the DIR of C-CNTs to a lower frequency zone. Originality/value To the best of our knowledge, analyses of anti-symmetric coupled CNTs have not received enough attentions so far. In order to optimize the nanostructures designing, the main purpose of the present paper is to investigate nonlocal dynamic stability of CNTs subjected to axial harmonic load coupled with CNTs conveying fluid.

Free vibration analysis of a rotating double tapered beam with flexible root via differential quadrature method

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mustafa Tolga Tolga Yavuz ◽  
İbrahim Özkol
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Operating Conditions ◽  
Design Parameters ◽  
Quadrature Method ◽  
Content Type ◽  
Uniform Beam ◽  
Governing Differential Equation

Purpose This study aims to develop the governing differential equation and to analyze the free vibration of a rotating non-uniform beam having a flexible root and setting angle for variations in operating conditions and structural design parameters. Design/methodology/approach Hamiltonian principle is used to derive the flapwise bending motion of the structure, and the governing differential equations are solved numerically by using differential quadrature with satisfactory accuracy and computation time. Findings The results obtained by using the differential quadrature method (DQM) are compared to results of previous studies in the open literature to show the power of the used method. Important results affecting the dynamics characteristics of a rotating beam are tabulated and illustrated in concerned figures to show the effect of investigated design parameters and operating conditions. Originality/value The principal novelty of this paper arises from the application of the DQM to a rotating non-uniform beam with flexible root and deriving new governing differential equation including various parameters such as rotary inertia, setting angle, taper ratios, root flexibility, hub radius and rotational speed. Also, the application of the used numerical method is expressed clearly step by step with the algorithm scheme.

Iterative differential quadrature algorithms for modified Burgers equation

2018 ◽  
Vol 35 (1) ◽  
pp. 235-250 ◽  
Author(s):  
Aswin V.S. ◽  
Ashish Awasthi
Keyword(s):  
Analytic Solution ◽  
Burgers Equation ◽  
Finite Difference Methods ◽  
Differential Quadrature ◽  
Mathematical Framework ◽  
Numerical Schemes ◽  
Content Type ◽  
Modified Burgers Equation ◽  
Highly Nonlinear ◽  
Grid Points

Purpose This paper aims to investigate the features of three vectorized iterative numerical schemes used to simulate the behavior of modified Burgers equation (MBE). Design/methodology/approach Two of the schemes comprise differential quadrature and finite difference methods, while the third scheme consists of only differential quadrature for the derivative approximations. Proposed schemes are simulated for well-posed problems of MBE having known the analytic solution. The computational complexity of the schemes is examined through monitoring the time taken to complete the simulation. The results are compared with the analytic solution with the help of discrete error norms. Also, the accuracy of the proposed schemes is compared with that of the existing schemes in the literature. Vectorized MATLAB programs of the schemes are used for all investigations. Findings It is observed that all the three schemes succeeded in producing a good replication of the exact solution. The results are closer to the analytical solution than the results in the literature. Among the three schemes, the scheme labeled as FDTDQS is found highly accurate and computationally cheaper using fewer grid points. From the vectorized MATLAB programs provided, it is evident that the implementation of the schemes is simple. Originality/value This study gives an idea about three numerical schemes for a highly nonlinear problem. This mathematical framework can be adopted to any one-dimensional partial differential equation as well, and the provided program will be helpful to generate more fast and accurate vectorized code in MATLAB.

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.

Sign in / Sign up

Export Citation Format

Share Document