Nonlinear coupled torsional-radial vibration of single-walled carbon nanotubes using numerical methods

Document Type : Research Paper

Authors

1 Faculty of Mechanical Engineering, Kar Higher Education Institute, Qazvin, Iran

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

3 Department of Mechanical Engineering, Shahryar Branch, Islamic Azad University, Shahryar, Iran

4 Department of Mechanical Engoneering, Shahid Chamran University of Ahwaz, Ahwaz, Iran

Abstract

This paper analyzes the nonlinear coupled torsional-radial vibration of single-walled carbon nanotubes (SWCNTs) based on numerical methods. Two partial differential equations that govern the nonlinear coupled torsional-radial vibration for such nanotube are derived using doublet mechanics (DM) principles. First, these equations are reduced to ordinary differential equations using Galerkin method and then solved using homotopy perturbation method (HPM) to obtain the nonlinear natural frequencies in coupled torsional-radial vibration mode. It is found that the obtained frequencies are complicated due to coupling between two vibration modes. The dependence of boundary conditions, vibration modes and nanotubes geometry on the nonlinear coupled torsional-radial vibration characteristics of SWCNTs are studied in details. It was shown that boundary conditions and maximum vibration velocity have significant effects on the nonlinear coupled torsional-radial vibration response of SWCNTs. It was also seen that unlike the linear model, as the maximum vibration velocity increases, the natural frequencies of vibration increase too. To show the effectiveness and ability of this method, the results obtained with the present method are compared with the fourth order Runge-Kuta numerical results and good agreement is observed. To the knowledge of authors, the results given herein are new and can be used as a basic work for future papers.

Keywords

[1]          A. Fatahi‐Vajari, "A new method for evaluating the natural frequency in radial breathing like mode vibration of double‐walled carbon nanotubes," ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 98, no. 2, pp. 255-269, 2018.
[2]          M. Ferrari, V. Granik, and A. Imam, "Introduction to doublet mechanics," in Advances in Doublet Mechanics: Springer, 1997, pp. 1-26.
[3]          M. Shishesaz, M. Shariati, and A. Yaghootian, "Nonlocal elasticity effect on linear vibration of nano-circular plate using adomian decomposition method," Journal of Applied and Computational Mechanics, vol. 6, no. 1, pp. 63-76, 2020.
[4]          M. Pourabdy, M. Shishesaz, S. Shahrooi, and S. A. S. Roknizadeh, "Analysis of Axisymmetric Vibration of Functionally-Graded ‎Circular Nano-Plate Based on the Integral Form of the Strain ‎Gradient Model," Journal of Applied and Computational Mechanics, pp. -, 2021, doi: 10.22055/jacm.2021.37461.3021.
[5]          A. Hadi, M. Z. Nejad, and M. Hosseini, "Vibrations of three-dimensionally graded nanobeams," International Journal of Engineering Science, vol. 128, pp. 12-23, 2018.
[6]          A. Hadi, M. Z. Nejad, A. Rastgoo, and M. Hosseini, "Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory," Steel and Composite Structures, vol. 26, no. 6, pp. 663-672, 2018.
[7]          M. Hosseini, H. H. Gorgani, M. Shishesaz, and A. Hadi, "Size-dependent stress analysis of single-wall carbon nanotube based on strain gradient theory," International Journal of Applied Mechanics, vol. 9, no. 06, p. 1750087, 2017.
[8]          M. Hosseini, A. Hadi, A. Malekshahi, and M. Shishesaz, "A review of size-dependent elasticity for nanostructures," Journal of Computational Applied Mechanics, vol. 49, no. 1, pp. 197-211, 2018.
[9]          M. Hosseini, M. Shishesaz, and A. Hadi, "Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness," Thin-Walled Structures, vol. 134, pp. 508-523, 2019.
[10]        M. Hosseini, M. Shishesaz, K. N. Tahan, and A. Hadi, "Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials," International Journal of Engineering Science, vol. 109, pp. 29-53, 2016.
[11]        M. M. Khoram, M. Hosseini, A. Hadi, and M. Shishehsaz, "Bending Analysis of Bidirectional FGM Timoshenko Nanobeam Subjected to Mechanical and Magnetic Forces and Resting on Winkler–Pasternak Foundation," International Journal of Applied Mechanics, vol. 12, no. 08, p. 2050093, 2020.
[12]        M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, and A. Rastgoo, "Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads," European Journal of Mechanics-A/Solids, vol. 77, p. 103793, 2019.
[13]        M. Mousavi Khoram, M. Hosseini, and M. Shishesaz, "A concise review of nano-plates," Journal of Computational Applied Mechanics, vol. 50, no. 2, pp. 420-429, 2019.
[14]        M. Shishesaz and M. Hosseini, "Mechanical behavior of functionally graded nano-cylinders under radial pressure based on strain gradient theory," Journal of Mechanics, vol. 35, no. 4, pp. 441-454, 2019.
[15]        M. Shishesaz, M. Hosseini, K. N. Tahan, and A. Hadi, "Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory," Acta Mechanica, vol. 228, no. 12, pp. 4141-4168, 2017.
[16]        S. M. Bachilo, M. S. Strano, C. Kittrell, R. H. Hauge, R. E. Smalley, and R. B. Weisman, "Structure-assigned optical spectra of single-walled carbon nanotubes," science, vol. 298, no. 5602, pp. 2361-2366, 2002.
[17]        A. M. Rao et al., "Diameter-selective Raman scattering from vibrational modes in carbon nanotubes," Science, vol. 275, no. 5297, pp. 187-191, 1997.
[18]        S. Gupta and R. Batra, "Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes," Computational Materials Science, vol. 43, no. 4, pp. 715-723, 2008.
[19]        S. Gupta, F. Bosco, and R. Batra, "Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and inextensional modes of vibration," Computational Materials Science, vol. 47, no. 4, pp. 1049-1059, 2010.
[20]        D. Sánchez-Portal, E. Artacho, J. M. Soler, A. Rubio, and P. Ordejón, "Ab initio structural, elastic, and vibrational properties of carbon nanotubes," Physical Review B, vol. 59, no. 19, pp. 12678-12688, 05/15/ 1999, doi: 10.1103/PhysRevB.59.12678.
[21]        M. Shariati, B. Azizi, M. Hosseini, and M. Shishesaz, "On the calibration of size parameters related to non-classical continuum theories using molecular dynamics simulations," International Journal of Engineering Science, vol. 168, p. 103544, 2021.
[22]        C. W. Lim, C. Li, and J. Yu, "Free torsional vibration of nanotubes based on nonlocal stress theory," Journal of Sound and Vibration, vol. 331, no. 12, pp. 2798-2808, 2012.
[23]        M. M. Adeli, A. Hadi, M. Hosseini, and H. H. Gorgani, "Torsional vibration of nano-cone based on nonlocal strain gradient elasticity theory," The European Physical Journal Plus, vol. 132, no. 9, pp. 1-10, 2017.
[24]        M. Shariati, M. Shishesaz, R. Mosalmani, S. A. Seyed Roknizadeh, and M. Hosseini, "Nonlocal effect on the axisymmetric nonlinear vibrational response of nano-disks using variational iteration method," Journal of Computational Applied Mechanics, vol. 52, no. 3, pp. 507-534, 2021.
[25]        M. Shariati, M. Shishesaz, H. Sahbafar, M. Pourabdy, and M. Hosseini, "A review on stress-driven nonlocal elasticity theory," Journal of Computational Applied Mechanics, vol. 52, no. 3, pp. 535-552, 2021.
[26]        M. Shishesaz, M. Shariati, and M. Hosseini, "Size effect analysis on Vibrational response of Functionally Graded annular nano plate based on Nonlocal stress-driven method," International Journal of Structural Stability and Dynamics, 2021, In press.
[27]        M. Shariati, M. Shishesaz, R. Mosalmani, and S. A. S Roknizadeh, "Size Effect on the Axisymmetric Vibrational Response of‎ Functionally Graded Circular Nano-Plate Based on the Nonlocal Stress-Driven Method," Journal of Applied and Computational Mechanics, 2021.
[28]        M. Ebrahimian, A. Imam, and M. Najafi, "Doublet mechanical analysis of bending of Euler‐Bernoulli and Timoshenko nanobeams," ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 98, no. 9, pp. 1642-1665, 2018.
[29]        M. Ebrahimian, A. Imam, and M. Najafi, "The effect of chirality on the torsion of nanotubes embedded in an elastic medium using doublet mechanics," Indian Journal of Physics, vol. 94, no. 1, pp. 31-45, 2020.
[30]        A. Fatahi-Vajari and A. Imam, "Axial vibration of single-walled carbon nanotubes using doublet mechanics," Indian Journal of Physics, vol. 90, no. 4, pp. 447-455, 2016.
[31]        A. Fatahi-Vajari and Z. Azimzadeh, "Analysis of nonlinear axial vibration of single-walled carbon nanotubes using Homotopy perturbation method," Indian Journal of Physics, vol. 92, no. 11, pp. 1425-1438, 2018.
[32]        S. Iijima, "Helical microtubules of graphitic carbon," nature, vol. 354, no. 6348, pp. 56-58, 1991.
[33]        A. Fatahi-Vajari and Z. Azimzadeh, "Axial vibration of single-walled carbon nanotubes with fractional damping using doublet mechanics," Indian Journal of Physics, vol. 94, no. 7, pp. 975-986, 2020.
[34]        R. Ansari, R. Gholami, and H. Rouhi, "Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories," Composites Part B: Engineering, vol. 43, no. 8, pp. 2985-2989, 2012.
[35]        I. Elishakoff and D. Pentaras, "Fundamental natural frequencies of double-walled carbon nanotubes," Journal of Sound and Vibration, vol. 322, no. 4-5, pp. 652-664, 2009.
[36]        A. Fatahi-Vajari and A. Imam, "Lateral vibrations of single-layered graphene sheets using doublet mechanics," Journal of Solid Mechanics, vol. 8, no. 4, pp. 875-894, 2016.
[37]        M. Arda and M. Aydogdu, "Analysis of free torsional vibration in carbon nanotubes embedded in a viscoelastic medium," Advances in Science and Technology. Research Journal, vol. 9, no. 26, 2015.
[38]        A. Fatahi-Vajari and A. Imam, "Torsional vibration of single-walled carbon nanotubes using doublet mechanics," Zeitschrift für angewandte Mathematik und Physik, vol. 67, no. 4, pp. 1-22, 2016.
[39]        M. Selim, "Torsional vibration of carbon nanotubes under initial compression stress," Brazilian Journal of Physics, vol. 40, pp. 283-287, 2010.
[40]        Q. Li and M. Shi, "Intermittent transformation between radial breathing and flexural vibration modes in a single-walled carbon nanotube," Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 464, no. 2096, pp. 1941-1953, 2008.
[41]        A. Fatahi‐Vajari and A. Imam, "Analysis of radial breathing mode of vibration of single‐walled carbon nanotubes via doublet mechanics," ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 96, no. 9, pp. 1020-1032, 2016.
[42]        M. Aydogdu, "Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity," Mechanics Research Communications, vol. 43, pp. 34-40, 2012.
[43]        S. Basirjafari, S. Esmaeilzadeh Khadem, and R. Malekfar, "Validation of shell theory for modeling the radial breathing mode of a single-walled carbon nanotube," Int. J. Eng. Trans. A, vol. 26, no. 4, pp. 447-454, 2013.
[44]        J. Maultzsch, H. Telg, S. Reich, and C. Thomsen, "Radial breathing mode of single-walled carbon nanotubes: Optical transition energies and chiral-index assignment," Physical Review B, vol. 72, no. 20, p. 205438, 2005.
[45]        S. Basirjafari, S. E. Khadem, and R. Malekfar, "Radial breathing mode frequencies of carbon nanotubes for determination of their diameters," Current Applied Physics, vol. 13, no. 3, pp. 599-609, 2013.
[46]        J.-H. He, "Approximate analytical solution for seepage flow with fractional derivatives in porous media," Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57-68, 1998.
[47]        J.-H. He, "Limit cycle and bifurcation of nonlinear problems," Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 827-833, 2005.
[48]        J.-H. He, "Homotopy perturbation technique," Computer methods in applied mechanics and engineering, vol. 178, no. 3-4, pp. 257-262, 1999.
[49]        J.-H. He, "Some asymptotic methods for strongly nonlinear equations," International journal of Modern physics B, vol. 20, no. 10, pp. 1141-1199, 2006.
[50]        M. Shishesaz, M. Shariati, A. Yaghootian, and A. Alizadeh, "Nonlinear Vibration Analysis of Nano-Disks Based on Nonlocal Elasticity Theory Using Homotopy Perturbation Method," International Journal of Applied Mechanics, vol. 11, no. 02, p. 1950011, 2019.
[51]        A. Vahidi, Z. Azimzadeh, and M. Didgar, "An efficient method for solving Riccati equation using homotopy perturbation method," Indian Journal of Physics, vol. 87, no. 5, pp. 447-454, 2013.
[52]        M. Ghasemia and K. M. TAVASSOLI, "Application of He’s homotopy perturbation method to solve a diffusion-convection problem," 2010.
[53]        Z. Azimzadeh, A. Vahidi, and E. Babolian, "Exact solutions for non-linear Duffing’s equations by He’s homotopy perturbation method," Indian Journal of Physics, vol. 86, no. 8, pp. 721-726, 2012.
[54]        M. Ghasemi, M. T. Kajani, and A. Davari, "Numerical solution of two-dimensional nonlinear differential equation by homotopy perturbation method," Applied Mathematics and Computation, vol. 189, no. 1, pp. 341-345, 2007.
[55]        A. P. Boresi, K. Chong, and J. D. Lee, Elasticity in engineering mechanics. John Wiley & Sons, 2010.
Volume 52, Issue 4
December 2021
Pages 642-663
  • Receive Date: 04 November 2021
  • Revise Date: 20 December 2021
  • Accept Date: 24 December 2021
  • First Publish Date: 24 December 2021