Skip to main content
SpringerLink
Log in

A simplified-nonlocal model for transverse vibration of nanotubes acted upon by a moving nanoparticle

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

This study provides a simplified solution for estimating the dynamic response of a single-walled carbon nanotube when excited by a moving nanoparticle. At first, the strong form of the equation of motion for a nonlocal Rayleigh nanotube is deduced, and the inertia effect of a moving nanoparticle along a nanobeam is then considered. For obtaining a weak form of the above nonlocal model, we use the Galerkin method, where the test functions are a set of orthogonal polynomials generated from a polynomial satisfying given boundary conditions. This process leads to a second-order differential equation which for a moving load the matrix coefficients are time dependent. In the state-space formulation, the forced response depends upon a transition matrix that can be locally approximated by the matrix exponential by assuming that the coefficients are locally constant. The normalized frequencies for a moving force are calculated and compared to those obtained in previous studies, and good agreement between them was observed. After acquiring the dynamic responses of a nanotube for a wide range of velocities and weights of moving nanoparticles, as well as for the nonlocal effects on a nanobeam, a nonlinear regression analysis is adapted to estimate the response of a nanobeam according to an analogous classical Rayleigh beam. These equivalent results in three multipliers (\(\alpha\), \(\beta\), and \(\gamma\)) are functions of kinetic parameters and nonlocal effects. Due to the normalization of the variables, these multipliers can be used for various types of beam-like structures in both the nano- and macro-domains. The accuracy of these coefficients is evaluated using the results gained by the analytical solution. This paper offers a remedy for a time-consuming process by means of some simple substitutions.

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

Similar content being viewed by others

References

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

    Article  Google Scholar 

  2. Bhirde AA, Patel V, Gavard J, Zhang G, Sousa AA, Masedunskas A, Leapman RD, Weigert R, Gutkind JS, Rusling JF (2009) Targeted killing of cancer cells in vivo and in vitro with EGF-directed carbon nanotube-based drug delivery. ACS Nano 3(2):307–316

    Article  Google Scholar 

  3. Hu R, Cola BA, Haram N, Barisci JN, Lee S, Stoughton S, Wallace G, Too C, Thomas M, Gestos A et al (2010) Harvesting waste thermal energy using a carbon-nanotube-based thermo-electrochemical cell. Nano Lett 10(3):838–846

    Article  Google Scholar 

  4. Pathangi H, Cherman V, Khaled A, Soree B, Groeseneken G, Witvrouw A (2013) Towards CMOS-compatible single-walled carbon nanotube resonators. Microelectron Eng 107:219–222

    Article  Google Scholar 

  5. Sazonova V, Yaish Y, Üstünel H, Roundy D, Arias TA, McEuen PL (2004) A tunable carbon nanotube electromechanical oscillator. Nature 431(7006):284–287

    Article  Google Scholar 

  6. Wang Q, Varadan V (2006) Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Mater Struct 15(2):659

    Article  Google Scholar 

  7. Reddy J, Pang S (2008) Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J Appl Phys 103(2):023511

    Article  Google Scholar 

  8. Eltaher M, Alshorbagy AE, Mahmoud F (2013a) Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Appl Math Model 37(7):4787–4797

    Article  MathSciNet  Google Scholar 

  9. Eltaher M, Mahmoud F, Assie A, Meletis E (2013b) Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams. Appl Math Comput 224:760–774

    MathSciNet  MATH  Google Scholar 

  10. Civalek Ö, Demir Ç (2011) Bending analysis of microtubules using nonlocal euler-bernoulli beam theory. Appl Math Model 35(5):2053–2067

    Article  MathSciNet  MATH  Google Scholar 

  11. Kiani K, Mehri B (2010) Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories. J Sound Vib 329(11):2241–2264

    Article  Google Scholar 

  12. Behera L, Chakraverty S (2014) Free vibration of Euler and Timoshenko nanobeams using boundary characteristic orthogonal polynomials. Appl Nanosci 4(3):347–358

    Article  MATH  Google Scholar 

  13. Şimşek M (2010) Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory. Phys E Low-Dimens Syst Nanostruct 43(1):182–191

    Article  Google Scholar 

  14. Şimşek M (2011) Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle. Comput Mater Sci 50(7):2112–2123

    Article  Google Scholar 

  15. Eltaher M, Hamed M, Sadoun A, Mansour A (2014) Mechanical analysis of higher order gradient nanobeams. Appl Math Comput 229:260–272

    MathSciNet  MATH  Google Scholar 

  16. Kiani K (2010) Longitudinal and transverse vibration of a single-walled carbon nanotube subjected to a moving nanoparticle accounting for both nonlocal and inertial effects. Phys E Low-Dimens Syst Nanostruct 42(9):2391–2401

    Article  Google Scholar 

  17. Kiani K (2014) Nonlinear vibrations of a single-walled carbon nanotube for delivering of nanoparticles. Nonlinear Dyn 76(4):1885–1903

    Article  MathSciNet  MATH  Google Scholar 

  18. Eringen AC (2002) Nonlocal continuum field theories. Springer Science & Business Media, USA

  19. Challamel N, Wang CM (2008) The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19(34):345703

    Article  Google Scholar 

  20. Fernandez-Saez J, Zaera R, Loya JA, Reddy JN (2016) Bending of Euler-Bernoulli beams using Eringens integral formulation: a paradox resolved. Int J Eng Sci 99:107–116

    Article  MathSciNet  Google Scholar 

  21. Kiani K (2016) Nonlocal-integro-differential modeling of vibration of elastically supported nanorods. Phys E Low-Dimens Syst Nanostruct 83:151–163

    Article  Google Scholar 

  22. Kiani K (2016) Free dynamic analysis of functionally graded tapered nanorods via a newly developed nonlocal surface energy-based integro-differential model. Compos Struct 139:151–166

    Article  MathSciNet  Google Scholar 

  23. Norouzzadeh A, Ansari R (2017) Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity. Phys E Low-Dimens Syst Nanostruct 88:194–200

    Article  Google Scholar 

  24. Norouzzadeh A, Ansari R, Rouhi H (2017) Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity: an isogeometric approach. Appl Phys A 123:330

    Article  Google Scholar 

  25. Steele C (1967) The finite beam with a moving load. J Appl Mech 34(1):111–118

    Article  MATH  Google Scholar 

  26. Knowles J (1968) On the dynamic response of a beam to a randomly moving load. J Appl Mech 35(1):1–6

    Article  MATH  Google Scholar 

  27. Hayashikawa T, Watanabe N (1981) Dynamic behavior of continuous beams with moving loads. J Eng Mech Div 107(1):229–246

    Google Scholar 

  28. Jaiswal O, Iyengar R (1993) Dynamic response of a beam on elastic foundation of finite depth under a moving force. Acta Mech 96(1–4):67–83

    Article  Google Scholar 

  29. Fryba L (1999) Vibration of solids and structures under moving loads. Thomas Telford, London

    Book  MATH  Google Scholar 

  30. Akin JE, Mofid M (1989) Numerical solution for response of beams with moving mass. J Struct Eng 115(1):120–131

    Article  Google Scholar 

  31. Ichikawa M, Miyakawa Y, Matsuda A (2000) Vibration analysis of the continuous beam subjected to a moving mass. J Sound Vib 230(3):493–506

    Article  Google Scholar 

  32. Michaltsos G, Kounadis A (2001) The effects of centripetal and Coriolis forces on the dynamic response of light bridges under moving loads. J Vib Control 7(3):315–326

    Article  MATH  Google Scholar 

  33. Nikkhoo A, Rofooei F, Shadnam M (2007) Dynamic behavior and modal control of beams under moving mass. J Sound Vib 306(3):712–724

    Article  MathSciNet  Google Scholar 

  34. Hasheminejad SM, Rafsanjani A (2011) Two-dimensional elasticity solution for transient response of simply supported beams under moving loads. Acta Mech 217(3–4):205–218

    Article  MATH  Google Scholar 

  35. Kiani K, Nikkhoo A (2012) On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method. Acta Mech Sin 28(1):164–179

    Article  MathSciNet  MATH  Google Scholar 

  36. Rajabi K, Kargarnovin M, Gharini M (2013) Dynamic analysis of a functionally graded simply supported Euler-Bernoulli beam subjected to a moving oscillator. Acta Mech 224(2):425–446

    Article  MathSciNet  MATH  Google Scholar 

  37. Pirmoradian M, Keshmiri M, Karimpour H (2014) On the parametric excitation of a Timoshenko beam due to intermittent passage of moving masses: instability and resonance analysis. Acta Mech 226(4):1241–1253

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang YM, Ko MY (2014) The interaction dynamics of a vehicle traveling along a simply supported beam under variable velocity condition. Acta Mech 225(12):3601–3616

    Article  MathSciNet  MATH  Google Scholar 

  39. Nikkhoo A (2014) Investigating the behavior of smart thin beams with piezoelectric actuators under dynamic loads. Mech Syst Signal Process 45(2):513–530

    Article  Google Scholar 

  40. Nikkhoo A, Farazandeh A, Hassanabadi ME, Mariani S (2015) Simplified modeling of beam vibrations induced by a moving mass by regression analysis. Acta Mech 226(7):2147–2157

    Article  MathSciNet  MATH  Google Scholar 

  41. Brogan WL (1991) Modern control theory, 3rd edn. Prentice Hall, USA

    MATH  Google Scholar 

  42. Nikkhoo A, Rofooei FR (2012) Parametric study of the dynamic response of thin rectangular plates traversed by a moving mass. Acta Mech 223(1):15–27

    Article  MathSciNet  MATH  Google Scholar 

  43. Kiani K, Wang Q (2012) On the interaction of a single-walled carbon nanotube with a moving nanoparticle using nonlocal Rayleigh, Timoshenko, and higher-order beam theories. Eur J Mech A/Solids 31(1):179–202

    Article  MathSciNet  MATH  Google Scholar 

  44. Wang C, Zhang Y, He X (2007) Vibration of nonlocal Timoshenko beams. Nanotechnology 18(10):105401

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, University of Science and Culture (USC), P.O. Box 13145-871, Tehran, Iran

    Ali Nikkhoo & Saber Zolfaghari

  2. Department of Civil Engineering, K.N.Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran

    Keivan Kiani

Authors
  1. Ali Nikkhoo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saber Zolfaghari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Keivan Kiani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Keivan Kiani.

Additional information

Technical Editor: Kátia Lucchesi Cavalca Dedini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nikkhoo, A., Zolfaghari, S. & Kiani, K. A simplified-nonlocal model for transverse vibration of nanotubes acted upon by a moving nanoparticle. J Braz. Soc. Mech. Sci. Eng. 39, 4929–4941 (2017). https://doi.org/10.1007/s40430-017-0892-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40430-017-0892-8

Keywords

Search

Navigation