Influence of various setting angles on vibration behavior of rotating graphene sheet: continuum modeling and molecular dynamics simulation

  • Amir Akbarshahi
  • Ali Rajabpour
  • Majid GhadiriEmail author
  • Mohammad Mostafa Barooti
Original Paper


The Eringen’s nonlocal elasticity theory is employed to examine the free vibration of a rotating cantilever single-layer graphene sheet (SLGS) under low and high temperature conditions. The governing equations of motion and the related boundary conditions are obtained through Hamilton’s principle based on the first-order shear deformation theory (FSDT) of nanoplates. The generalized differential quadrature method (GDQM) is utilized to solve the nondimensional equations of motion. The molecular dynamics (MD) simulation is conducted, and fundamental frequencies of the rotating cantilever square SLGS are computed using the fast Fourier transform (FFT). The comparison of MD and GDQM results leads to finding the appropriate value of the nonlocal parameter for the first time. As an interesting result, this value of the nonlocal parameter is independent of the angular velocity. Results indicate that increases in various parameters, such as the angular velocity, hub radius, nonlocal parameter, and temperature changes in low temperature conditions, leads the first and the second frequencies to increase. In addition, it can be seen that the influence of the hub radius or nonlocal parameters on frequencies cannot be ignored in high angular velocities. Moreover, it is not possible to neglect the angular velocity or nonlocal parameter in high hub radius. The results show that the influence of parameters such as setting angle or nonlocal parameter on the first and the second frequencies increases when some parameters increase, such as the angular velocity, hub radius or temperature change.

Graphical abstract

(a) A schematic of a rotating cantilever nanoplate. (b) A schematic of cantilever armchair SLGS simulated by MD.


Vibration analysis Rotating nanoplate Thermal effect Eringen’s nonlocal theory Molecular dynamics simulation GDQM 



  1. 1.
    Roy S, Gao Z (2009) Nanostructure-based electrical biosensors. Nano Today 4(4):318–334CrossRefGoogle Scholar
  2. 2.
    Kuilla T et al (2010) Recent advances in graphene based polymer composites. Prog Polym Sci 35(11):1350–1375CrossRefGoogle Scholar
  3. 3.
    Li X et al (2009) Integrated MEMS/NEMS resonant cantilevers for ultrasensitive biological detection. J Sensors. Scholar
  4. 4.
    Bunch JS et al (2007) Electromechanical resonators from graphene sheets. Science 315(5811):490–493PubMedCrossRefGoogle Scholar
  5. 5.
    Ji Y et al (2012) Organic nonvolatile memory devices with charge trapping multilayer graphene film. Nanotechnology 23(10):105202PubMedCrossRefGoogle Scholar
  6. 6.
    Pradhan S, Murmu T (2010) Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory. Phys E 42(5):1293–1301CrossRefGoogle Scholar
  7. 7.
    Wang X, Zhi L, Müllen K (2008) Transparent, conductive graphene electrodes for dye-sensitized solar cells. Nano Lett 8(1):323–327PubMedCrossRefGoogle Scholar
  8. 8.
    Verre S, Ombres L, Politano A (2017) Evaluation of the free-vibration frequency and the variation of the bending rigidity of graphene nanoplates: the role of the shape geometry and boundary conditions. J Nanosci Nanotechnol 17(12):8827–8834CrossRefGoogle Scholar
  9. 9.
    Fadaee M (2016) Buckling analysis of a defective annular graphene sheet in elastic medium. Appl Math Model 40(3):1863–1872CrossRefGoogle Scholar
  10. 10.
    Eringen AC (2002) Nonlocal continuum field theories. Springer, BerlinGoogle Scholar
  11. 11.
    Eringen AC, Edelen D (1972) On nonlocal elasticity. Int J Eng Sci 10(3):233–248CrossRefGoogle Scholar
  12. 12.
    Ghadiri M, Rajabpour A, Akbarshahi A (2018) Non-linear vibration and resonance analysis of graphene sheet subjected to moving load on a visco-Pasternak foundation under thermo-magnetic-mechanical loads: an analytical and simulation study. Measurement 124:103–119CrossRefGoogle Scholar
  13. 13.
    Ghadiri M, Rajabpour A, Akbarshahi A (2017) Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects. Appl Math Model 50:676–694CrossRefGoogle Scholar
  14. 14.
    Liu J et al (2017) Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory. Appl Math Model 45:65–84CrossRefGoogle Scholar
  15. 15.
    Liew K, Zhang Y, Zhang L (2017) Nonlocal elasticity theory for graphene modeling and simulation: prospects and challenges. J Model Mech Mater 1:1–7Google Scholar
  16. 16.
    Muraoka T, Kinbara K, Aida T (2006) Mechanical twisting of a guest by a photoresponsive host. Nature 440(7083):512–515PubMedCrossRefGoogle Scholar
  17. 17.
    Serreli V et al (2007) A molecular information ratchet. Nature 445(7127):523–527PubMedCrossRefGoogle Scholar
  18. 18.
    Liu Y et al (2005) Linear artificial molecular muscles. J Am Chem Soc 127(27):9745–9759PubMedCrossRefGoogle Scholar
  19. 19.
    Li J et al (2014) Rotation motion of designed nano-turbine. Sci Rep 4:5846Google Scholar
  20. 20.
    Bedard TC, Moore JS (1995) Design and synthesis of molecular turnstiles. J Am Chem Soc 117(43):10662–10671CrossRefGoogle Scholar
  21. 21.
    Wang L, Wu H, Wang F (2017) Design of nano screw pump for water transport and its mechanisms. Sci Rep 7:41717Google Scholar
  22. 22.
    Tu Q et al (2016) Rotating carbon nanotube membrane filter for water desalination. Sci Rep 6:26183PubMedPubMedCentralCrossRefGoogle Scholar
  23. 23.
    Rao J (2011) Evolution of rotor dynamics in 20th century. In: World Congress in Mechanism and Machine Science, Guanajuato, MexicoGoogle Scholar
  24. 24.
    Genta G (2007) Dynamics of rotating systems. Springer, BerlinGoogle Scholar
  25. 25.
    Wang J (2012) Cargo-towing synthetic nanomachines: towards active transport in microchip devices. Lab Chip 12(11):1944–1950PubMedCrossRefGoogle Scholar
  26. 26.
    Kim K et al (2014) Ultrahigh-speed rotating nanoelectromechanical system devices assembled from nanoscale building blocks. Nat Commun 5:3632PubMedCrossRefGoogle Scholar
  27. 27.
    Ghalichechian N et al (2008) Design, fabrication, and characterization of a rotary micromotor supported on microball bearings. J Microelectromech Syst 17(3):632–642CrossRefGoogle Scholar
  28. 28.
    Frechette LG et al (2001) An electrostatic induction micromotor supported on gas-lubricated bearings. In: Micro Electro Mechanical Systems, 2001. The 14th IEEE International ConferenceGoogle Scholar
  29. 29.
    Cook E, et al (2015) Fabrication of a rotary carbon nanotube bearing test apparatus. In: Journal of Physics: Conference Series. IOP, BristolGoogle Scholar
  30. 30.
    Southwell R, Gough F (1921) The free transverse vibration of airscrew blades. British ARC Reports and Memoranda No. 766Google Scholar
  31. 31.
    Schilhansl M (1958) Bending frequency of a rotating cantilever beam. J Appl Mech 25:28–30Google Scholar
  32. 32.
    Popplewell N, Chang D (1997) Free vibrations of a stepped, spinning Timoshenko beam. J Sound Vib 203(4):717–722CrossRefGoogle Scholar
  33. 33.
    Yu S, Cleghorn W (2000) Free vibration of a spinning stepped Timoshenko beam. J Appl Mech 67(4):839–841CrossRefGoogle Scholar
  34. 34.
    Lin S, Hsiao K (2001) Vibration analysis of a rotating Timoshenko beam. J Sound Vib 240(2):303–322CrossRefGoogle Scholar
  35. 35.
    Ghadiri M, Shafiei N, Akbarshahi A (2016) Influence of thermal and surface effects on vibration behavior of nonlocal rotating Timoshenko nanobeam. Applied Physics A 7(122):1–19Google Scholar
  36. 36.
    Ghadiri M, Shafiei N (2016) Vibration analysis of rotating functionally graded Timoshenko microbeam based on modified couple stress theory under different temperature distributions. Acta Astronaut 121:221–240CrossRefGoogle Scholar
  37. 37.
    Ehyaei J, Akbarshahi A, Shafiei N (2017) Influence of porosity and axial preload on vibration behavior of rotating FG nanobeam. Adv Nano Res 5(2):141–169Google Scholar
  38. 38.
    Ghadiri M, Shafiei N (2016) Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method. Microsyst. Technol. 22(12):2853–2867CrossRefGoogle Scholar
  39. 39.
    Bediz B, Romero L, Ozdoganlar OB (2015) Three dimensional dynamics of rotating structures under mixed boundary conditions. J Sound Vib 358:176–191CrossRefGoogle Scholar
  40. 40.
    Yoo H, Kim S (2002) Flapwise bending vibration of rotating plates. Int J Numer Methods Eng 55(7):785–802CrossRefGoogle Scholar
  41. 41.
    Hashemi S, Farhadi S, Carra S (2009) Free vibration analysis of rotating thick plates. J Sound Vib 323(1):366–384CrossRefGoogle Scholar
  42. 42.
    Fang J, Zhou D (2017) Free vibration analysis of rotating mindlin plates with variable thickness. Int J Struct Stab Dyn 17(04):1750046CrossRefGoogle Scholar
  43. 43.
    Yoo HH, Kim SK (2002) Free vibration analysis of rotating cantilever plates. AIAA J 40(11):2188–2196CrossRefGoogle Scholar
  44. 44.
    Yoo H, Pierre C (2003) Modal characteristic of a rotating rectangular cantilever plate. J Sound Vib 259(1):81–96CrossRefGoogle Scholar
  45. 45.
    Dokainish M, Rawtani S (1971) Vibration analysis of rotating cantilever plates. Int J Numer Methods Eng 3(2):233–248CrossRefGoogle Scholar
  46. 46.
    Ramamurti V, Kielb R (1984) Natural frequencies of twisted rotating plates. J Sound Vib 97(3):429–449CrossRefGoogle Scholar
  47. 47.
    Joseph SV, Mohanty S (2017) Free vibration of a rotating Sandwich plate with viscoelastic core and functionally graded material constraining layer. Int J Struct Stab Dyn. Scholar
  48. 48.
    Hamza-Cherif SM (2006) Free vibration analysis of rotating cantilever plates using the p-version of the finite element method. Struct Eng Mech 22(2):151–167CrossRefGoogle Scholar
  49. 49.
    Kou H, Yuan H (2014) Rub-induced non-linear vibrations of a rotating large deflection plate. Int J Non-Linear Mech 58:283–294CrossRefGoogle Scholar
  50. 50.
    Eisenberger M, Deutsch A (2015) Static analysis for exact vibration analysis of clamped plates. Int J Struct Stab Dyn 15(08):1540030CrossRefGoogle Scholar
  51. 51.
    Ruocco E, Minutolo V, Ciaramella S (2011) A generalized analytical approach for the buckling analysis of thin rectangular plates with arbitrary boundary conditions. Int J Struct Stab Dyn 11(01):1–21CrossRefGoogle Scholar
  52. 52.
    Civalek Ö (2009) Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method. Appl Math Model 33(10):3825–3835CrossRefGoogle Scholar
  53. 53.
    Mishra I, Sahu SK (2015) Modal analysis of woven fiber composite plates with different boundary conditions. Int J Struct Stab Dyn 15(01):1540001CrossRefGoogle Scholar
  54. 54.
    Huang B-W (2004) The drilling vibration behavior of a twisted microdrill. Trans ASME-B J Manuf Sci Eng 126(4):719–726CrossRefGoogle Scholar
  55. 55.
    Huang Y-M, Lee C-Y (1998) Dynamics of a rotating rayleigh beam subject to a repetitively travelling force. Int J Mech Sci 40(8):779–792CrossRefGoogle Scholar
  56. 56.
    Lennard-Jones JE, Strachan C (1935) The interaction of atoms and molecules with solid surfaces. I. The activation of adsorbed atoms to higher vibrational states. Proc R Soc Lond A Math Phys Sci 150(870):442–455Google Scholar
  57. 57.
    Tersoff J (1989) Modeling solid-state chemistry: interatomic potentials for multicomponent systems. Phys Rev B 39(8):5566CrossRefGoogle Scholar
  58. 58.
    Brenner DW (1990) Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys Rev B 42(15):9458CrossRefGoogle Scholar
  59. 59.
    Stuart SJ, Tutein AB, Harrison JA (2000) A reactive potential for hydrocarbons with intermolecular interactions. J Chem Phys 112(14):6472–6486CrossRefGoogle Scholar
  60. 60.
    Sahmani S, Fattahi A (2017) Development an efficient calibrated nonlocal plate model for nonlinear axial instability of zirconia nanosheets using molecular dynamics simulation. J Mol Graph Model 75:20–31PubMedCrossRefGoogle Scholar
  61. 61.
    Ansari R, Sahmani S, Arash B (2010) Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys Lett A 375(1):53–62CrossRefGoogle Scholar
  62. 62.
    Pishkenari HN, Afsharmanesh B, Akbari E (2015) Surface elasticity and size effect on the vibrational behavior of silicon nanoresonators. Curr Appl Phys 15(11):1389–1396CrossRefGoogle Scholar
  63. 63.
    Hashemnia K, Farid M, Vatankhah R (2009) Vibrational analysis of carbon nanotubes and graphene sheets using molecular structural mechanics approach. Comput Mater Sci 47(1):79–85CrossRefGoogle Scholar
  64. 64.
    Shakouri A, Ng T, Lin R (2013) A study of the scale effects on the flexural vibration of graphene sheets using REBO potential based atomistic structural and nonlocal couple stress thin plate models. Phys E 50:22–28CrossRefGoogle Scholar
  65. 65.
    Sadeghi M, Naghdabadi R (2010) Nonlinear vibrational analysis of single-layer graphene sheets. Nanotechnology 21(10):105705PubMedCrossRefGoogle Scholar
  66. 66.
    Kang JW, Lee S (2013) Molecular dynamics study on the bending rigidity of graphene nanoribbons. Comput Mater Sci 74:107–113CrossRefGoogle Scholar
  67. 67.
    Kwon OK et al (2013) Developing ultrasensitive pressure sensor based on graphene nanoribbon: molecular dynamics simulation. Phys E 47:6–11CrossRefGoogle Scholar
  68. 68.
    Bellman R, Casti J (1971) Differential quadrature and long-term integration. J Math Anal Appl 34(2):235–238CrossRefGoogle Scholar
  69. 69.
    Shu C (2012) Differential quadrature and its application in engineering. Springer, BerlinGoogle Scholar
  70. 70.
    Leissa AW (1969) Vibration of plates. Ohio State Univ Columbus, ColumbusGoogle Scholar
  71. 71.
    Wang J-S, Shaw D, Mahrenholtz O (1987) Vibration of rotating rectangular plates. J Sound Vib 112(3):455–468CrossRefGoogle Scholar
  72. 72.
    Shafiei N et al (2017) Thermo-mechanical vibration analysis of rotating nonlocal nanoplates applying generalized differential quadrature method. Mech Adv Mater Struct 24(15):1257–1273Google Scholar
  73. 73.
    Thai H-T et al (2014) A nonlocal sinusoidal plate model for micro/nanoscale plates. Proc Inst Mech Eng C J Mech Eng Sci 228(14):2652–2660CrossRefGoogle Scholar
  74. 74.
    Liu C-C, Chen Z-B (2014) Dynamic analysis of finite periodic nanoplate structures with various boundaries. Phys E 60:139–146CrossRefGoogle Scholar
  75. 75.
    Ansari R, Rouhi H (2013) An explicit nonlocal frequency formula for monolayer graphene sheets. Int J Comput Methods Eng Sci Mech 14(1):40–44CrossRefGoogle Scholar
  76. 76.
    Arash B, Wang Q (2011) Vibration of single-and double-layered graphene sheets. J Nanotechnol Eng Med 2(1):011012CrossRefGoogle Scholar
  77. 77.
    Nazemnezhad R (2015) Nonlocal Timoshenko beam model for considering shear effect of van der Waals interactions on free vibration of multilayer graphene nanoribbons. Compos Struct 133:522–528CrossRefGoogle Scholar
  78. 78.
    Ansari R, Ajori S (2014) Molecular dynamics study of the torsional vibration characteristics of boron-nitride nanotubes. Phys Lett A 378(38-39):2876–2880CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Amir Akbarshahi
    • 1
  • Ali Rajabpour
    • 1
  • Majid Ghadiri
    • 1
    Email author
  • Mohammad Mostafa Barooti
    • 1
  1. 1.Advanced Simulation and Computing Laboratory, Mechanical Engineering DepartmentImam Khomeini International UniversityQazvinIran

Personalised recommendations