International Journal of Differential Equations

Nonlinear Vibrations of Multiwalled Carbon Nanotubes under Various Boundary Conditions

Hossein Aminikhah and Milad Hemmatnezhad

Full-text: Access by subscription

Abstract

The present work deals with applying the homotopy perturbation method to the problem of the nonlinear oscillations of multiwalled carbon nanotubes embedded in an elastic medium under various boundary conditions. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the van der Waals interlayer forces. The amplitude-frequency curves for large-amplitude vibrations of single-walled, double-walled, and triple-walled carbon nanotubes are obtained. The influences of some commonly used boundary conditions, changes in material constant of the surrounding elastic medium, and variations of the nanotubes geometrical parameters on the vibration characteristics of multiwalled carbon nanotubes are discussed. The comparison of the generated results with those from the open literature illustrates that the solutions obtained are of very high accuracy and clarifies the capability and the simplicity of the present method. It is worthwhile to say that the results generated are new and can be served as a benchmark for future works.

Article information

Source
Int. J. Differ. Equ., Volume 2011 (2011), Article ID 343576, 17 pages.

Dates
Received: 7 May 2011
Accepted: 20 June 2011
First available in Project Euclid: 25 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485313237

Digital Object Identifier
doi:10.1155/2011/343576

Mathematical Reviews number (MathSciNet)
MR2843504

Zentralblatt MATH identifier
1235.82108

Citation

Aminikhah, Hossein; Hemmatnezhad, Milad. Nonlinear Vibrations of Multiwalled Carbon Nanotubes under Various Boundary Conditions. Int. J. Differ. Equ. 2011 (2011), Article ID 343576, 17 pages. doi:10.1155/2011/343576. https://projecteuclid.org/euclid.ijde/1485313237


Export citation

References

  • S. Iijima, “Helical microtubules of graphitic carbon,” Nature, vol. 354, no. 6348, pp. 56–58, 1991.
  • C. Q. Ru and K. M. Liew, “Elastic models for carbon nanotubes,” in Encyclopedia of Nanoscience and Nanotechnology, vol. 4, pp. 1–14, 2003.
  • S. Adali, “Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory,” Physics Letters, Section A, vol. 372, no. 35, pp. 5701–5705, 2008.
  • J. N. Ding, B. Kan, G. G. Cheng, Z. Fan, N. Y. Yuan, and Z. Y. Ling, “Numerical approach to torsion deformation of armchair single walled carbon nanotubes,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 4, pp. 309–314, 2008.
  • J. H. He, “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering,” International Journal of Modern Physics B, vol. 22, no. 21, pp. 3487–3578, 2008.
  • I. Elishakoff and D. Pentaras, “Fundamental natural frequencies of double-walled carbon nanotubes,” Journal of Sound and Vibration, vol. 322, no. 4-5, pp. 652–664, 2009.
  • C. Q. Ru, “Intrinsic vibration of multiwalled carbon nanotubes,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 3, no. 3-4, p. 735, 2002.
  • S. B. Ye, R. P. S. Han, and L. H. Wang, “Oscillatory response of a capped double-walled carbon nanotube,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 4, pp. 339–346, 2008.
  • J. Yoon, C. Q. Ru, and A. Mioduchowski, “Noncoaxial resonance of an isolated multiwall carbon nanotube,” Physical Review B, vol. 66, no. 23, Article ID 233402, pp. 2334021–2334024, 2002.
  • J. Yoon, C. Q. Ru, and A. Mioduchowski, “Vibration of an embedded multiwall carbon nanotube,” Composites Science and Technology, vol. 63, no. 11, pp. 1533–1542, 2003.
  • Y. Zhang, G. Liu, and X. Han, “Transverse vibrations of double-walled carbon nanotubes under compressive axial load,” Physics Letters, Section A, vol. 340, no. 1–4, pp. 258–266, 2005.
  • Y. M. Fu, J. W. Hong, and X. Q. Wang, “Analysis of nonlinear vibration for embedded carbon nanotubes,” Journal of Sound and Vibration, vol. 296, no. 4-5, pp. 746–756, 2006.
  • J. H. He, “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004.
  • J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 695–700, 2005.
  • J. H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters, Section A, vol. 350, no. 1-2, pp. 87–88, 2006.
  • J. H. He, “Limit cycle and bifurcation of nonlinear problems,” Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 827–833, 2005.
  • J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999.
  • J. H. He, “Coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000.
  • J. H. He, “Comparison of homotopy perturbation method and homotopy analysis method,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 527–539, 2004.
  • J. H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003.
  • J. H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006.
  • D. D. Ganji, “The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer,” Physics Letters, Section A, vol. 355, no. 4-5, pp. 337–341, 2006.
  • D. D. Ganji and A. Sadighi, “Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 24–34, 2007.
  • A. Rajabi, D. D. Ganji, and H. Taherian, “Application of homotopy perturbation method in nonlinear heat conduction and convection equations,” Physics Letters, Section A, vol. 360, no. 4-5, pp. 570–573, 2007.
  • S. Abbasbandy, “A numerical solution of Blasius equation by Adomian's decomposition method and comparison with homotopy perturbation method,” Chaos, Solitons & Fractals, vol. 31, no. 1, pp. 257–260, 2007.
  • J. Biazar and H. Ghazvini, “Exact solutions for non-linear Schrödinger equations by He's homotopy perturbation method,” Physics Letters, Section A, vol. 366, no. 1-2, pp. 79–84, 2007.
  • L. Cveticanin, “Application of homotopy-perturbation to non-linear partial differential equations,” Chaos, Solitons & Fractals, vol. 40, no. 1, pp. 221–228, 2009.
  • A. Y. T. Leung and Z. Guo, “Homotopy perturbation for conservative Helmholtz-Duffing oscillators,” Journal of Sound and Vibration, vol. 325, no. 1-2, pp. 287–296, 2009.
  • S. Abbasbandy, “Numerical solutions of the integral equations: homotopy perturbation method and Adomian's decomposition method,” Applied Mathematics and Computation, vol. 173, no. 1, pp. 493–500, 2006.
  • J.-H. He, “Recent development of the homotopy perturbation method,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 205–209, 2008.
  • J. Biazar and H. Aminikhah, “Study of convergence of homotopy perturbation method for systems of partial differential equations,” Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2221–2230, 2009.
  • H. T. Hahn and J. G. Williams, “Compression failure mechanisms in unidirectional composites,” Composite Materials, vol. 7, pp. 115–139, 1984.
  • Y. Lanir and Y. C. B. Fung, “Fiber composite columns under compression,” Journal of Composite Materials, vol. 6, pp. 387–402, 1972.
  • F.S. Tse, I. E. Morse, and R. T. Hinkle, Mechanical Vibrations: Theory and Applications, Allyn and Bacon Inc., Boston, Mass, USA, 2nd edition, 1978.
  • R. Ansari, M. Hemmatnezhad, and H. Ramezannezhad, “Application of HPM to the nonlinear vibrations ofmultiwalled carbon nanotubesčommentComment on ref. [4?]: Please update the information of this reference, if possible.,” Numerical Methods for Partial Differential Equations, vol. 26, pp. 490–500, 2010.