Skip to main content
SpringerLink
Log in

Vibration analysis of viscoelastic single-walled carbon nanotubes resting on a viscoelastic foundation

  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

Vibration responses were investigated for a viscoelastic Single-walled carbon nanotube (visco-SWCNT) resting on a viscoelastic foundation. Based on the nonlocal Euler-Bernoulli beam model, velocity-dependent external damping and Kelvin viscoelastic foundation model, the governing equations were derived. The Transfer function method (TFM) was then used to compute the natural frequencies for general boundary conditions and foundations. In particular, the exact analytical expressions of both complex natural frequencies and critical viscoelastic parameters were obtained for the Kelvin-Voigt visco-SWCNTs with full foundations and certain boundary conditions, and several physically intuitive special cases were discussed. Substantial nonlocal effects, the influence of geometric and physical parameters of the SWCNT and the viscoelastic foundation were observed for the natural frequencies of the supported SWCNTs. The study demonstrates the efficiency and robustness of the developed model for the vibration of the visco-SWCNT-viscoelastic foundation coupling system.

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

Similar content being viewed by others

References

  1. S. Iijima, Helical microtubules of graphitic carbon, Nature, 354 (1991) 56–58.

    Article  Google Scholar 

  2. A. Shahsavar, M. R. Salimpour, M. Saghafian and M. B. Shafii, Effect of magnetic field on thermal conductivity and viscosity of a magnetic nanofluid loaded with carbon nanotubes, J. of Mechanical Science and Technology, 30 (2016) 809–815.

    Article  Google Scholar 

  3. T. Murmu, M. A. McCarthy and S. Adhikari, Vibration response of double-walled carbon nanotubes subjected to an externally applied longitudinal magnetic field: A nonlocal elasticity approach, J. of Sound and Vibration, 331 (2012) 5069–5086.

    Article  Google Scholar 

  4. N. Wattanasakulpong and V. Ungbhakorn, Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation, Composite Materials Science, 71 (2013) 201–208.

    MATH  Google Scholar 

  5. R. Syed, W. Jiang, C. Wang and M. I. Sabir, Fatigue life of stainless steel 304 enhancement by addition of multi-walled carbon nanotubes (MWCNTs), J. of Mechanical Science and Technology, 29 (2015) 291–296.

    Article  Google Scholar 

  6. A. Azrar, L. Azrar and A. A. Aljinaidi, Numerical modeling of dynamic and parametric instabilities of single-walled carbon nanotubes conveying pulsating and viscous fluid, Composite Structures, 125 (2015) 127–143.

    Article  Google Scholar 

  7. Z. B. Shen, X. F. Li, L. P. Sheng and G. J. Tang, Transverse vibration of nanotube-based micro-mass sensor via nonlocal Timoshenko beam theory, Computational Materials Science, 53 (2012) 340–346.

    Article  Google Scholar 

  8. Z. B. Shen, G. J. Tang, L. Zhang and X. F. Li, Vibration of double-walled carbon nanotube based nanomechanical sensor with initial axial stress, Computational Materials Science, 58 (2012) 51–58.

    Article  Google Scholar 

  9. H. L. Tang, Z. B. Shen and D. K. Li, Vibration of nonuniform carbon nanotube with attached mass via nonlocal Timoshenko beam theory, J. of Mechanical Science and Technology, 28 (2014) 3741–3747.

    Article  Google Scholar 

  10. H. M. Sedighi and F. Daneshmand, Static and dynamic pull-in instability of multi-walled carbon nanotube probes by He's iteration perturbation method, J. of Mechanical Science and Technology, 28 (2014) 3459–3469.

    Article  Google Scholar 

  11. M. A. Torkaman-Asadi, M. Rahmanian and R. D. Firouz-Abadi, Free vibrations and stability of high-speed rotating carbon nanotubes partially resting on Winkler foundations, Composite Structures, 126 (2015) 52–61.

    Article  Google Scholar 

  12. M. A. Kazemi-Lari, S. A. Fazelzadeh and E. Ghavanloo, Non-conservative instability of cantilever carbon nanotubes resting on viscoelastic foundation, Physica E, 44 (2012) 1623–1630.

    Article  Google Scholar 

  13. I. Mehdipour, A. Barari, A. Kimiaeifar and G. Domairry, Vibrational analysis of curved single-walled carbon nanotube on a Pasternak elastic foundation, Advances in Engineering Software, 48 (2012) 1–5.

    Article  Google Scholar 

  14. K. Kiani, Vibration analysis of elastically restrained double-walled carbon nanotubes on elastic foundation subjected to axial load using nonlocal shear deformable beam theories, International J. of Mechanical Science, 68 (2013) 16–34.

    Article  Google Scholar 

  15. E. Ghavanloo, F. Daneshmand and M. Rafiei, Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation, Physica E, 42 (2010) 2218–2224.

    Article  Google Scholar 

  16. Z. B. Shen, D. K. Li, D. Li and G. J. Tang, Frequency shift of a nanomechanical sensor carrying a nanopartical using nonlocal Timoshenko theory, J. of Mechanical Science and Technology, 26 (2012) 1577–1583.

    Article  Google Scholar 

  17. K. B. Mustapha and Z. W. Zhong, The thermo-mechanical vibraton of a single-walled carbon nanotube studied using the Bubnov-Galerkin method, Physica E, 43 (2010) 375–381.

    Article  Google Scholar 

  18. S. Adhikari, D. Gilchrist, T. Murmu and M. A. McCarthy, Nonlocal normal modes in nanoscale dynamical systems, Mechanical Systems and Signal Processing, 60-61 (2015) 583–603.

    Article  Google Scholar 

  19. T.-P. Chang, Small scale effect on axial vibration of nonuniform and non-homogeneous nanorods, Computational Materials Science, 54 (2012) 23–27.

    Article  Google Scholar 

  20. Y. Lei, S. Adhikari and M. I. Friswell, Vibration of nonlocal Kelvin-Voigt viscoelastic damped Timoshenko beams, International J. of Engineering Science, 66-67 (2013) 1–13.

    Article  MathSciNet  Google Scholar 

  21. Z. B. Shen, H. L. Tang, D. K. Li and G. J. Tang, Vibration of single-layered graphene sheet-based nanomechanical sensor via nonlocal Kirchhoff plate theory, Computational Materials Science, 61 (2012) 200–205.

    Article  Google Scholar 

  22. F. Ebrahimi and E. Salari, Thermo-mechanical vibration analysis of a single-walled carbon nanotube embedded in an elastic medium based on higher-order shear deformation beam theory, J. of Mechanical Science and Technology, 29 (2015) 3797–3803.

    Article  Google Scholar 

  23. A. C. Eringen, On differential equations of nonlocal elasticity and solution of screw dislocation and surface waves, J. of Applied Physics, 54 (1983) 4703–4710.

    Article  Google Scholar 

  24. A. C. Eringen, A unified continuum theory of electrodynamics of liquid crystals, International J. of Engineering Science, 35 (1997) 1137–1157.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. C. Eringen, Theory of nonlocal pasticity, International J. of Engineering Science, 21 (1983) 741–751.

    Article  MATH  Google Scholar 

  26. J. Peddieson, G. R. Buchanan and R. P. McNitt, Application of nonlocal continuum models to nanotechnology, International J. of Engineering Science, 41 (2003) 305–312.

    Article  Google Scholar 

  27. L. J. Sudak, Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics, J. of Applied Physics, 94 (2003) 72–81.

    Article  Google Scholar 

  28. J. Reddy, Nonlocal theories for bending, bucking and vibration of beams, International J. of Engineering Science, 45 (2007) 288–307.

    MATH  Google Scholar 

  29. Y. Lei, S. Adhikari, T. Murmu and M. I. Friswell, Asymptotic frequencies of various damped nonlocal beams and plates, Mechanics Research Communications, 62 (2014) 94–101.

    Article  Google Scholar 

  30. Y. Lei, T. Murmu, S. Adhikari and M. I. Friswell, Dynamic characteristics of damped viscoelastic nonlocal Euler-Bernoulli beams, European J. of Mechanics A/Solids, 42 (2013) 125–136.

    Article  MathSciNet  Google Scholar 

  31. A. Ghasemi, M. Dardel, M. H. Ghasemi and M. M. Barzegari, Analytical analysis of buckling and post-buckling of fluid conveying multi-walled carbon nanotubes, Applied Mathematical Modelling, 37 (2013) 4972–4992.

    Article  MathSciNet  Google Scholar 

  32. R. Ansari and H. Ramezannezhad, Nonlocal Timoshenko beam model for the large-amplitude vibrations of embedded multiwalled carbon nanotubes including thermal effects, Physica E, 43 (2011) 1171–1178.

    Article  Google Scholar 

  33. M. S. Hoseinzadeh and S. E. Khadem, A nonlocal shell theory model for evaluation of thermoelastic damping in the vibration of a double-walled carbon nanotube, Physica E, 57 (2014) 6–11.

    Article  Google Scholar 

  34. R. Ansari, H. Rouhi and S. Sahmani, Calibration of the analytical nonlocal shell model for vibrations of doublewalled carbon nanotubes with arbitrary boundary conditions using molecular dynamics, International J. of Mechanical Sciences, 53 (2011) 786–792.

    Article  Google Scholar 

  35. P. Soltani and A. Farshidianfar, Periodic solution for nonlinear vibration of a fluid-conveying carbon nanotube, based on the nonlocal continuum theory by energy balance method, Applied Mathematical Modelling, 36 (2012) 3712–3724.

    MATH  Google Scholar 

  36. D. Thamviratnam and Y. Zhuge, Free vibration analysis of beams on elastic foundation, Computers and Structures, 60 (1996) 971–980.

    Article  MATH  Google Scholar 

  37. Y. Lei, Finite element analysis of beams with nonlocal foundations, 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conferrence, Newport, Rhode Island (2006) 1-11.

    Google Scholar 

  38. M. I. Friswell, S. Adhikari and Y. Lei, Vibration analysis of beams with non-local foundations using the finite element method, International J. for Numerical Methods in Engineering, 71 (2007) 1365–1386.

    Article  MATH  Google Scholar 

  39. C. P. Wu and W. W. Lai, Reissner’s mixed variational theorem-based nonlocal Timoshenko beam theory for a single-walled carbon nanotube embedded in an elastic medium and with various boundary conditions, Composite Structures, 122 (2015) 390–404.

    Article  Google Scholar 

  40. M. M. Fotouhi, R. D. Firouz-Abadi and H. Haddadpour, Free vibration analysis of nanocones embedded in an elastic medium using a nonlocal continuum shell model, International J. of Engineering Science, 64 (2013) 14–22.

    Article  MathSciNet  Google Scholar 

  41. H. Zeighampour and Y. T. Beni, Size-dependent vibration of fluid-conveying double-walled carbon nanotubes using couple stress shell theory, Physica E, 61 (2014) 28–39.

    Article  Google Scholar 

  42. C. A. Cooper, R. J. Young and M. Halsall, Investigation into the deformation of carbon nanotuves and their composites through the use of Raman spectroscopy, Composites: Part A, 32 (2001) 401–411.

    Article  Google Scholar 

  43. P. Soltani, M. M. Taherian and A. Farshidianfar, Vibration and instability of a viscous-fluid-conveying single-walled carbon nanotube embedded in a visco-elastic medium, J. of Physics D: Applied Physics, 43 (2010) 425401.

    Article  Google Scholar 

  44. B. Arash and Q. Wang, A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Computational Materials Science, 51 (2012) 303–313.

    Article  Google Scholar 

  45. J. X. Huang, M. F. Song, L. Zhang, P. Liu, R. X. Chen, J. H. He and S. Q. Wang, Transverse vibration of an axially moving slender fiber of viscoelastic fluid in bubbfil spinning and stuffer box crimping, Thermal Science, 19 (2015) 1437–1441.

    Article  Google Scholar 

  46. P. Lu, H. P. Lee, C. Lu and P. Q. Zhang, Dynamic properties of flexural beams using a nonlocal elasticity model, J. of Applied Physics, 99 (2006) 073510.

    Article  Google Scholar 

  47. C. M. Wang, Y. Y. Zhang and X. Q. He, Vibration of nonlocal Timoshenko beams, Nanotechnology, 18 (2007) 105401.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, Hunan, 410073, China

    Da-Peng Zhang, Yong-Jun Lei & Zhi-Bin Shen

  2. Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Swansea Wales, SA2 8PP, UK

    Cheng-Yuan Wang

Authors
  1. Da-Peng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong-Jun Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cheng-Yuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhi-Bin Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong-Jun Lei.

Additional information

Recommended by Associate Editor Ohseop Song

Da-Peng Zhang obtained his M.S. from the National University of Defense Technology (NUDT), China in 2013. Currently, he is a Ph.D. candidate of the College of Aerospace Science and Engineering at the NUDT. His research interests include vibration analyses of carbon nanotubes and graphene sheets.

Yong-Jun Lei received his Ph.D. in solid mechanics at the National University of Defense Technology (NUDT), China in 1998. Dr. Lei is currently a Professor in the College of Aerospace Science and Engineering at the NUDT. His research interests include theoretical and applied computational solid mechanics.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, DP., Lei, YJ., Wang, CY. et al. Vibration analysis of viscoelastic single-walled carbon nanotubes resting on a viscoelastic foundation. J Mech Sci Technol 31, 87–98 (2017). https://doi.org/10.1007/s12206-016-1007-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-016-1007-7

Keywords

Search

Navigation