A new method for evaluating the natural frequency in radial breathing like mode vibration of double‐walled carbon nanotubes

Fatahi-Vajari, Alireza

doi:10.1002/zamm.201600234

Cited by 7 publications

(4 citation statements)

References 50 publications

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“…3–6 to give the frequencies in THz. Throughout this Letter, the mechanical properties of SWCNT are assumed to be: Poisson's ratio

ν = 0.2

, mass density

ρ = 2300 false(kg / m^{3} false)

and Young's modulus

E = 1.1 normalTPa

[1].…”

Section: Resultsmentioning

confidence: 99%

“…In the cylindrical coordinates, the equations of motion are [1, 24, 25]

\frac{normal∂ N_{z z}}{normal∂ z} + \frac{1}{r} \frac{normal∂ N_{θ z}}{normal∂ θ} + ρ f_{z} = ρ \frac{{normal∂}^{2} u_{z}}{\partial t^{2}}

\frac{normal∂ N_{z θ}}{normal∂ z} + \frac{1}{r} \frac{normal∂ N_{θ θ}}{normal∂ θ} + \frac{N_{θ r}}{r} + ρ f_{θ} = ρ \frac{\partial^{2} u_{θ}}{normal∂ t^{2}}

\frac{normal∂ N_{z r}}{normal∂ z} + \frac{1}{r} \frac{normal∂ N_{θ r}}{normal∂ θ} - \frac{N_{θ θ}}{r} + ρ f_{r} = ρ \frac{\partial^{2} u_{r}}{normal∂ t^{2}}

\frac{normal∂ M_{z z}}{normal∂ z} + \frac{1}{r} \frac{normal∂ M_{θ z}}{normal∂ θ} + ρ {\overset{false¯}{l}}_{z} = N_{z r}

…”

Section: Basic Equations Of Motionmentioning

confidence: 99%

“…If distribution is continuous, theories are based on classical continuum mechanics (CCM) and do not contain any scaling effects. This feature normally is the most important limitation of CCM [1, 2]. On account of nanoscale dimensions of carbon nanotubes (CNTs), it is hard to implement accurate experiments to obtain the properties of a CNT [3].…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear coupled axial–torsional vibration of single‐walled carbon nanotubes using homotopy perturbation method

Fatahi-Vajari¹,

Azimzadeh²,

Hussain³

2019

Micro & Nano Letters

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This work analyses the nonlinear coupled axial-torsional vibration of single-walled carbon nanotubes (SWCNTs) based on numerical methods. Two-second order partial differential equations that govern the nonlinear coupled axial-torsional vibration for such nanotube are derived. First, these equations are reduced to ordinary differential equations using the Galerkin method and then solved using homotopy perturbation method (HPM) to obtain the nonlinear natural frequencies in coupled axial-torsional vibration mode. It is found that the obtained frequencies are complicated due to coupling between two vibration modes. The dependence of boundary conditions, vibration modes and nanotubes geometry on the nonlinear coupled axial-torsional vibration characteristics of SWCNTs are studied in detail. It was shown that boundary conditions and maximum initial vibration velocity have significant effects on the nonlinear coupled axial-torsional vibration response of SWCNTs. It was also seen that unlike the linear model if the maximum vibration velocity increases, the natural frequencies of vibration increases too. To show the effectiveness and ability of this method, the results obtained with HPM are compared with the fourth-order Runge-Kutta numerical results and good agreement is observed. To the knowledge of authors, the results given herein are new and can be used as a foundation work for future work.

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“…3–6 to give the frequencies in THz. Throughout this Letter, the mechanical properties of SWCNT are assumed to be: Poisson's ratio

ν = 0.2

, mass density

ρ = 2300 false(kg / m^{3} false)

and Young's modulus

E = 1.1 normalTPa

[1].…”

Section: Resultsmentioning

confidence: 99%

“…In the cylindrical coordinates, the equations of motion are [1, 24, 25]

\frac{normal∂ N_{z z}}{normal∂ z} + \frac{1}{r} \frac{normal∂ N_{θ z}}{normal∂ θ} + ρ f_{z} = ρ \frac{{normal∂}^{2} u_{z}}{\partial t^{2}}

\frac{normal∂ N_{z θ}}{normal∂ z} + \frac{1}{r} \frac{normal∂ N_{θ θ}}{normal∂ θ} + \frac{N_{θ r}}{r} + ρ f_{θ} = ρ \frac{\partial^{2} u_{θ}}{normal∂ t^{2}}

\frac{normal∂ N_{z r}}{normal∂ z} + \frac{1}{r} \frac{normal∂ N_{θ r}}{normal∂ θ} - \frac{N_{θ θ}}{r} + ρ f_{r} = ρ \frac{\partial^{2} u_{r}}{normal∂ t^{2}}

\frac{normal∂ M_{z z}}{normal∂ z} + \frac{1}{r} \frac{normal∂ M_{θ z}}{normal∂ θ} + ρ {\overset{false¯}{l}}_{z} = N_{z r}

…”

Section: Basic Equations Of Motionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Nonlinear coupled axial–torsional vibration of single‐walled carbon nanotubes using homotopy perturbation method

Fatahi-Vajari¹,

Azimzadeh²,

Hussain³

2019

Micro & Nano Letters

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show abstract

“…However, with an increase in the radius of SWCNTs, the influence of the scale parameter on the natural frequency decreases. Vajari [303] continued their study to investigate the radial breathing like mode vibration of DWCNTs based on the size-dependent DM. He assumed DWCNTs as two concentric cylindrical shells restrained together with Len-Jones potential modeled vdW forces.…”

Section: Nonlocal Doublet Mechanicsmentioning

confidence: 99%

Advances in modelling and analysis of nano structures: a review

Chandel

Wang

Talha

2020

Nanotechnology Reviews

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AbstractNanostructures are widely used in nano and micro-sized systems and devices such as biosensors, nano actuators, nano-probes, and nano-electro-mechanical systems. The complete understanding of the mechanical behavior of nanostructures is crucial for the design of nanodevices and systems. Therefore, the flexural, stability and vibration analysis of various nanostructures such as nanowires, nanotubes, nanobeams, nanoplates, graphene sheets and nanoshells has received a great attention in recent years. The focus has been made, to present the structural analysis of nanostructures under thermo-magneto-electro-mechanical loadings under various boundary and environmental conditions. This paper also provides an overview of analytical modeling methods, fabrication procedures, key challenges and future scopes of development in the direction of analysis of such structures, which will be helpful for appropriate design and analysis of nanodevices for the application in the various fields of nanotechnology.

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A Softening–Hardening Nanomechanics Theory for the Static and Dynamic Analyses of Nanorods and Nanobeams: Doublet Mechanics

Gül¹,

Aydoğdu²

2021

Modern Trends in Structural and Solid Mechanics 2

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A new method for evaluating the natural frequency in radial breathing like mode vibration of double‐walled carbon nanotubes

Cited by 7 publications

References 50 publications

Nonlinear coupled axial–torsional vibration of single‐walled carbon nanotubes using homotopy perturbation method

Nonlinear coupled axial–torsional vibration of single‐walled carbon nanotubes using homotopy perturbation method

Advances in modelling and analysis of nano structures: a review

A Softening–Hardening Nanomechanics Theory for the Static and Dynamic Analyses of Nanorods and Nanobeams: Doublet Mechanics

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