Skip to main content
SpringerLink
Log in

Analysis of nonlinear axial vibration of single-walled carbon nanotubes using Homotopy perturbation method

  • Original Paper
  • Published:
Indian Journal of Physics Aims and scope Submit manuscript

Abstract

This paper investigates the nonlinear axial vibration of single-walled carbon nanotubes (SWCNTs) based on Homotopy perturbation method (HPM). A second order partial differential equation that governs the nonlinear axial vibration for such nanotubes is derived using doublet mechanics (DM) theory. To obtain the nonlinear natural frequency in axial vibration mode, this nonlinear equation is solved using HPM. The influences of some commonly used boundary conditions, amplitude of vibration, changes in vibration modes and variations of the nanotubes geometrical parameters on the nonlinear axial vibration characteristics of SWCNTs are discussed. It was shown that unlike the linear one, the nonlinear natural frequency is dependent to maximum vibration amplitude. Increasing the maximum vibration amplitude decreases the natural frequency of vibration compared to the predictions of the linear models. However, with increase in tube length, the effect of the amplitude on the natural frequency decreases. It was also shown that the amount and variation of nonlinear natural frequency is more apparent in higher mode vibration and two clamped boundary conditions. To show the accuracy and capability of this method, the results obtained herein were compared with the fourth order Runge–Kuta numerical results and good agreement was observed. It is notable that the results generated herein are new and can be served as a benchmark for future works.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. J M Hill and A P S Selvadurai J. Eng. Math. 52 1 (2005)

    Article  Google Scholar 

  2. V T Granik and M Ferrari Mech. Mater. 15 301 (1993)

    Article  Google Scholar 

  3. S Forest and D K Trinh ZAMM. 91(2) 90 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  4. V A Eremeyev and L P Lebedev ZAMM. 91(6) 468 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  5. A Fatahi-Vajari and A Imam J. Solid Mech. 8(4) 875 (2016)

    Google Scholar 

  6. M Ferretti, A Madeo, F Dell’Isola and P Boisse ZAMP 65(3) 587 (2014)

    ADS  MathSciNet  Google Scholar 

  7. M Ferrari, V T Granik, A Imam and J Nadeau Advances in Doublet Mechanics (Berlin: Springer-Verlag) p 17 (1997)

  8. A Fatahi-Vajari and A Imam ZAMM. 96(9) 1020 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  9. M H Sadd and Q Dai Mech. Mater. 37 641 (2005)

    Article  Google Scholar 

  10. A Fatahi-Vajari and A Imam ZAMP. 67 https://doi.org/10.1007/s00033-016-0675-6 (2016)

  11. F Gentile, J Sakamoto, R Righetti, P Decuzzi and M Ferrari J. Biomed. Sci. Eng. 4 362 (2011)

    Article  Google Scholar 

  12. J Y Fang, Z Jue, F Jing and M Ferrari Chin. Phys. Lett. 21(8) 1562 (2004)

    Article  ADS  Google Scholar 

  13. A Fatahi-Vajari and A Imam Indian J. Phys. 90(4) 447 (2016)

    Article  ADS  Google Scholar 

  14. A Fatahi-Vajari ZAMM. 98(2) 255 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  15. A C Eringen Int. J. Eng. Sci. 10 1 (1972)

    Article  Google Scholar 

  16. F Dell’Isola, U Andreaus and L Placidi Math. Mech. Solids. 20(8) 887 (2014)

    Article  Google Scholar 

  17. F Dell ‘Isola, I Giorgio, M Pawlikowski and Nicola Rizzi Proceedings of the Royal Society of London A. 472(2185) 20150790 (2016)

  18. F Dell’Isola, A Della Corte and I Giorgio Math. Mech. Solids 22(4) 852 (2017)

    Article  MathSciNet  Google Scholar 

  19. F Dell’Isola, P Seppecher and A Madeo ZAMP. 63 1119 (2012)

    ADS  Google Scholar 

  20. Y Yang and A Misra Int. J. Solids Struct. 49(18) 2500 (2012)

    Article  Google Scholar 

  21. C S Chang, A Misra and S S Sundaram Soil Dyn. Earthq. Eng. 10(4) 201 (1991)

    Article  Google Scholar 

  22. A Misra and P Poorsolhjouy Contin. Mech. Thermodyn. 28(1–2) 215 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  23. G Ivan, A DellaCorte and F Dell’Isola Nonlinear Dyn. 88(1) 21 (2017)

    Article  Google Scholar 

  24. H Aminpour and N Rizzi Gen. Contin. Mod. Class. Adv. Mater. 42 15 (2016)

    Google Scholar 

  25. L Placidi, L Greco, S Bucci, E Turco and N L Rizzi ZAMP. 67(5) 114 (2016)

    ADS  Google Scholar 

  26. S S Gupta and R C Batra Comput. Mater. Sci.. 43 715 (2008)

    Article  Google Scholar 

  27. F S S Gupta, G Bosco and R C Batra Comput. Mater. Sci. 47 1049 (2010)

    Article  Google Scholar 

  28. J H He Comput. Meth. Appl. Mech. Eng. 167 57 (1998)

    Article  ADS  Google Scholar 

  29. J H He Chaos, Solitons Fract. 26(3) 695 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  30. J H He Phys. Lett. 350(1–2) 87 (2006)

    Article  MathSciNet  Google Scholar 

  31. J H He Int. J. Nonlinear Sci. Numer. Simul. 6 207 (2005)

  32. J H He Indian J. Phys. 88(2) 193 (2014)

    Article  ADS  Google Scholar 

  33. J H He Int. J. Mod. Phys. B. 20(10) 1141 (2006)

    Article  ADS  Google Scholar 

  34. M Ghasemia and M Tavassoli Kajani Math. Sci. 4(2) 171 (2010)

    MathSciNet  Google Scholar 

  35. B Raftari and A Yildirim Comput. Math. Appl. 59 3328 (2010)

    Article  MathSciNet  Google Scholar 

  36. Z Azimzadeh, A R Vahidi and E Babolian Indian J. Phys. 86(8) 721 (2012)

    Article  ADS  Google Scholar 

  37. S Abbasbandy Chaos, Solitons Fract. 31(1) 257 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  38. J Biazar and H Ghazvini Phys. Lett. 366(1–2) 79 (2007)

    Article  MathSciNet  Google Scholar 

  39. L Cveticanin Chaos. Solitons Fract. 40(1) 221 (2009)

  40. SS Nourazar and A Nazari-Golshan, Indian J. Phys. 89(1) 61 (2015)

    Article  ADS  Google Scholar 

  41. A Y T Leung and Z Guo J. Sound Vib. 325(1–2) 287 (2009)

    Article  ADS  Google Scholar 

  42. A Rajabi Phys. Lett. A. 364 33 (2007)

  43. A R Vahidi, Z Azimzadeh and M Didgar Indian J. Phys. 87(5) 447 (2013)

    Google Scholar 

  44. F Bouyge, I Jasiuk, S Boccara and M Ostoja-Starzewski Eur. J. Mech. A Solids 21 465 (2002)

    Article  Google Scholar 

  45. A P Boresi and K P Chong Elasticity in Engineering Mechanics (New York: John Wiley and Sons) p 334 (2000)

  46. S S Rao Mechanical Vibrations (Massachusetts: Addison-Wesley publishing company) p 383 (2000)

  47. G Kirchhoff Vorlesungen über Mathematische Physik. Erster Band. Mechanik (Leipzig: B.G.Teubner) p 53 (1897)

  48. A H Nayfeh Perturbation Methods (New York: John Wiley and Sons) p 159 (2000)

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Shahryar Branch, Islamic Azad University, Shahryar, Iran

    A. Fatahi-Vajari

  2. Department of Mathematics, College of Sciences, Yadegar-e-Imam Khomeini (RAH) Shahre Rey Branch, Islamic Azad University, Tehran, Iran

    Z. Azimzadeh

Authors
  1. A. Fatahi-Vajari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Z. Azimzadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Fatahi-Vajari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fatahi-Vajari, A., Azimzadeh, Z. Analysis of nonlinear axial vibration of single-walled carbon nanotubes using Homotopy perturbation method. Indian J Phys 92, 1425–1438 (2018). https://doi.org/10.1007/s12648-018-1203-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12648-018-1203-7

Keywords

PACS Nos.

Search

Navigation