Free vibration analysis of a rotationally restrained (FG) nanotube

  • Mustafa Özgür YayliEmail author
Technical Paper


In this study, free lateral vibration behavior of a functionally graded nanobeam in an elastic matrix with rotationally restrained ends is studied based on the Eringens’ nonlocal theory of elasticity formulated in differential form. Euler–Bernoulli beam theory, Fourier sine series and Stokes’ transformation are used to investigate the vibrational behavior of nanobeams with restrained boundary conditions. Although vibration based dynamical analysis of nanostructures is a widely investigated topic, there are only few studies that exist in the literature pertaining to the analysis of nanobeams with rotationally restrained boundary conditions. To investigate and analyze the effect of deformable boundary conditions on the lateral vibration of nanobeams, the Fourier coefficients obtained by using Stokes’ transformation. Explicit formulas are derived for the elastic nonlocal boundary conditions at the ends. A useful coefficient matrix is derived by using these equations. Moreover, the effects of some parameters such as functional gradient index, nonlocal parameter, and rotational restraints on the natural frequencies are studied and some conclusions are drawn.



  1. Ajayan PM, Lijima S (1992) Smallest carbon nanotube. Nature 358:23CrossRefGoogle Scholar
  2. Akbas SD (2018) Forced vibration analysis of cracked functionally graded microbeams. Adv Nano Res 6(1):39–55MathSciNetGoogle Scholar
  3. Antonelli GA, Maris HJ, Malhotra SG, Harper JM (2002) Picosecond ultrasonics study of the vibrational modes of a nanostructure. J Appl Phys 91(5):3261–3267CrossRefGoogle Scholar
  4. Arash B, Wang Q (2012) A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput Mater Sci 51(1):303–313CrossRefGoogle Scholar
  5. Arda M, Aydogdu M (2014) Torsional statics and dynamics of nanotubes embedded in an elastic medium. Compos Struct 114:80–91CrossRefGoogle Scholar
  6. Arroyo M, Belytschko T (2005) Continuum mechanics modeling and simulation of carbon nanotubes. Meccanica 40(4–6):455–469MathSciNetCrossRefzbMATHGoogle Scholar
  7. Aydogdu M (2009) Axial vibration of the nanorods with the nonlocal continuum rod model. Phys E Low Dimens Syst Nanostruct 41(5):861–864CrossRefGoogle Scholar
  8. Bachtold A, Hadley P, Nakanishi T, Dekker C (2001) Logic circuits with carbon nanotube transistors. Science 294(5545):1317–1320CrossRefGoogle Scholar
  9. Barretta R, Marotti de Sciarra F (2013) A nonlocal model for carbon nanotubes under axial loads. Adv Mater Sci Eng. Google Scholar
  10. Barretta R, Brcic M, Canadija M, Luciano R, de Sciarra FM (2017) Application of gradient elasticity to armchair carbon nanotubes: size effects and constitutive parameters assessment. Eur J Mech A Solids 65:1–13MathSciNetCrossRefzbMATHGoogle Scholar
  11. Brauns EB, Madaras ML, Coleman RS, Murphy CJ, Berg MA (2002) Complex local dynamics in DNA on the picosecond and nanosecond time scales. Phys Rev Lett 88(15):158101CrossRefGoogle Scholar
  12. Bunch JS, Van Der Zande AM, Verbridge SS, Frank IW, Tanenbaum DM, Parpia JM, Craighead HG, McEuen PL (2007) Electromechanical resonators from graphene sheets. Science 315(5811):490–493CrossRefGoogle Scholar
  13. Chien WT, Chen CS, Chen HH (2006) Resonant frequency analysis of fixed-free single-walled carbon nanotube-based mass sensor. Sens Actuators A Phys 126(1):117–121MathSciNetCrossRefGoogle Scholar
  14. Chiu HY, Hung P, Postma HWC, Bockrath M (2008) Atomic-scale mass sensing using carbon nanotube resonators. Nano Lett 8(12):4342–4346CrossRefGoogle Scholar
  15. Chong ACM, Lam DCC (1999) Strain gradient plasticity effect in indentation hardness of polymers. J Mater Res 14:4103–4110CrossRefGoogle Scholar
  16. Chowdhury R, Adhikari S, Mitchell J (2009) Vibrating carbon nanotube based bio-sensors. Phys E Low Dimens Syst Nanostruct 42(2):104–109CrossRefGoogle Scholar
  17. Dai H, Hafner JH, Rinzler AG, Colbert DT, Smalley RE (1996) Nanotubes as nanoprobes in scanning probe microscopy. Nature 384(6605):147CrossRefGoogle Scholar
  18. Demir C, Civalek O, Akgoz B (2010) Free vibration analysis of carbon nanotubes based on shear deformable beam theory by discrete singular convolution technique. Math Comput Appl 15(1):57–65zbMATHGoogle Scholar
  19. Ebrahimi F, Mahmoodi F (2018) Vibration analysis of carbon nanotubes with multiple cracks in thermal environment. Adv Nano Res 6(1):57–80Google Scholar
  20. El-Borgi S, Rajendran P, Friswell MI, Trabelssi M, Reddy JN (2018) Torsional vibration of size-dependent viscoelastic rods using nonlocal strain and velocity gradient theory. Compos Struct 186:274–292CrossRefGoogle Scholar
  21. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710CrossRefGoogle Scholar
  22. Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248MathSciNetCrossRefzbMATHGoogle Scholar
  23. Filiz S, Aydogdu M (2010) Axial vibration of carbon nanotube heterojunctions using nonlocal elasticity. Comput Mater Sci 49(3):619–627CrossRefGoogle Scholar
  24. Heireche H, Tounsi A, Benzair A, Mechab I (2008) Sound wave propagation in single-walled carbon nanotubes with initial axial stress. J Appl Phys 104(1):014301CrossRefGoogle Scholar
  25. Joshi AY, Harsha SP, Sharma SC (2010) Vibration signature analysis of single walled carbon nanotube based nanomechanical sensors. Phys E Low Dimens Syst Nanostruct 42(8):2115–2123CrossRefGoogle Scholar
  26. Kim P, Lieber CM (1999) Nanotube nanotweezers. Science 286(5447):2148–2150CrossRefGoogle Scholar
  27. Lau KT, Gu C, Hui D (2006) A critical review on nanotube and nanotube/nanoclay related polymer composite materials. Compos Part B Eng 37(6):425–436CrossRefGoogle Scholar
  28. Lim CW, Yang Y (2010) New predictions of size-dependent nanoscale based on nonlocal elasticity for wave propagation in carbon nanotubes. J Comput Theor Nanosci 7(6):988–995CrossRefGoogle Scholar
  29. Lu P, Lee HP, Lu C, Zhang PQ (2007) Application of nonlocal beam models for carbon nanotubes. Int J Solids Struct 44(16):5289–5300CrossRefzbMATHGoogle Scholar
  30. Mehdipour I, Barari A (2012) Why the center-point of bridged carbon nanotube length is the most mass sensitive location for mass attachment? Comput Mater Sci 55:136–141CrossRefGoogle Scholar
  31. Murmu T, Adhikari S, Wang CY (2011) Torsional vibration of carbon nanotube–buckyball systems based on nonlocal elasticity theory. Phys E Low Dimens Syst Nanostruct 43(6):1276–1280CrossRefGoogle Scholar
  32. Peddieson J, Buchanan GR, McNitt RP (2003) Application of nonlocal continuum models to nanotechnology. Int J Eng Sci 41(3–5):305–312CrossRefGoogle Scholar
  33. Pradhan SC, Murmu T (2009) Differential quadrature method for vibration analysis of beam on Winkler foundation based on nonlocal elastic theory. J Inst Eng (India) Metall Mater Eng Div 89:3–12Google Scholar
  34. Qian D, Wagner GJ, Liu WK, Yu MF, Ruoff RS (2002) Mechanics of carbon nanotubes. Appl Mech Rev 55(6):495–533CrossRefGoogle Scholar
  35. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45(2–8):288–307CrossRefzbMATHGoogle Scholar
  36. Schedin F, Geim AK, Morozov SV, Hill EW, Blake P, Katsnelson MI, Novoselov KS (2007) Detection of individual gas molecules adsorbed on graphene. Nature materials 6(9):652CrossRefGoogle Scholar
  37. Sirtori C (2002) Applied physics: bridge for the terahertz gap. Nature 417(6885):132CrossRefGoogle Scholar
  38. Stankovich S, Dikin DA, Dommett GH, Kohlhaas KM, Zimney EJ, Stach EA, Piner ST, Nguyen RS, Ruoff RS (2006) Graphene-based composite materials. Nature 442(7100):282CrossRefGoogle Scholar
  39. Sudak LJ (2003) Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J Appl Phys 94(11):7281–7287CrossRefGoogle Scholar
  40. Thostenson ET, Ren Z, Chou TW (2001) Advances in the science and technology of carbon nanotubes and their composites: a review. Compos Sci Technol 61(13):1899–1912CrossRefGoogle Scholar
  41. Wagner HD, Lourie O, Feldman Y, Tenne R (1998) Stress induced fragmentation of multiwall carbon nanotubes in a polymer matrix. Appl Phys Lett 72(2):188–190CrossRefGoogle Scholar
  42. Wan H, Delale F (2010) A structural mechanics approach for predicting the mechanical properties of carbon nanotubes. Meccanica 45(1):43–51CrossRefzbMATHGoogle Scholar
  43. Yayli MO (2016) A compact analytical method for vibration analysis of single-walled carbon nanotubes with restrained boundary conditions. J Vib Control 22(10):2542–2555MathSciNetCrossRefGoogle Scholar
  44. Yayli MO (2018a) An efficient solution method for the longitudinal vibration of nanorods with arbitrary boundary conditions via a hardening nonlocal approach. J Vib Control 24(11):2230–2246MathSciNetCrossRefGoogle Scholar
  45. Yayli MÖ (2018b) Torsional vibrations of restrained nanotubes using modified couple stress theory. Microsyst Technol 24(8):3425–3435CrossRefGoogle Scholar
  46. Yayli MÖ (2018c) On the torsional vibrations of restrained nanotubes embedded in an elastic medium. J Braz Soc Mech Sci Eng 40(9):419CrossRefGoogle Scholar
  47. Yayli MÖ (2018d) Free vibration analysis of a single-walled carbon nanotube embedded in an elastic matrix under rotational restraints. Micro Nano Lett 13(2):202–206MathSciNetCrossRefGoogle Scholar
  48. Yayli MÖ, Cercevik AE (2015) 1725. Axial vibration analysis of cracked nanorods with arbitrary boundary conditions. J Vibroeng 17(6):2907–2921Google Scholar
  49. Zhang YQ, Liu GR, Wang JS (2004) Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression. Phys Rev B 70(20):205430CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Engineering, Department of Civil EngineeringBursa Uludag UniversityBursaTurkey

Personalised recommendations