Natural Frequency of Rotating Single-Walled Carbon Nanotubes with Considering Gyroscopic Effect

Document Type: Research Paper


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

2 Young Researchers and Elite Club, Yadegar-e-Imam Khomeini (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran



This paper investigates the bending vibration of rotating single-walled carbon nanotubes (SWCNTs) based on nonlocal theory. To this end, the rotating SWCNTs system modeled as a beam with a circular cross section and the Euler-Bernoulli beam theory (EBT) is applied with added effects such as rotary inertia, gyroscopic effect and rotor mass unbalance. Using nonlocal theory, two coupled sixth order partial differential equations that govern the vibration of rotating SWCNTs are derived. To obtain the natural frequency and dynamic response of the nanorotor system, the equation of motion for the rotating SWCNTs are solved. It is found that there are two frequencies in the frequency spectrum. The positive rootintroduced as forward whirling mode, while the negative root represents backward whirling mode. The detailed mathematical derivations are presented while the emphasis is placed on investigating the effect of the several parameters such as, tube radius, angular velocity and small scale parameter on the vibration behavior of rotating nanotubes. It is explicitly shown that the vibration of a spinning nanotube is significantly influenced by these effects. To validate the accuracy and efficiency of this work, the results obtained herein are compared with the existing theoretical and experimental results and good agreement is observed. To the knowledge of authors, the vibration of rotating SWCNTs considering gyroscopic effect has not investigated analytically yet and then the results generated herein can be served as a benchmark for future works.


[1] Iijima S., 1991, Helical microtubes of graphitic carbon, Nature 354: 56-58.
[2] Basirjafari S., Khadem S. E., Malekfar R., 2013, Radial breathing mode frequencies of carbon nanotubes for determination of their diameters, Current Applied Physics 13: 599-609.
[3] Fatahi-Vajari A., 2018, A new method for evaluating the natural frequency in radial breathing like mode vibration of double-walled carbon nanotubes, ZAMM 98(2): 255-269.
[4] Rao S. S., 2000, Mechanical Vibrations, Addison-Wesley Publishing Company, Massachusetts.
[5] Gupta S. S., Bosco F. G., Batra R. C., 2010, Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and in extensional modes of vibration, Computational Materials Science 47: 1049-1059.
[6] Fatahi-Vajari A., Imam A., 2016, Torsional vibration of single-walled carbon nanotubes using doublet mechanics, ZAMP 67: 81.
[7] Ferrari M., Granik V. T., Imam A., Nadeau J., 1997, Advances in Doublet Mechanics, Springer, Berlin.
[8] Fatahi-Vajari A., Imam A., 2016, Analysis of radial breathing mode of vibration of single-walled carbon nanotubes via doublet mechanics, ZAMM 96(9): 1020-1032.
[9] Eringen A. C., 1972, Nonlocal polar elastic, International Journal of Engineering Science 10: l-16.
[10] Basirjafari S., Esmaielzadeh Khadem S., Malekfar R., 2013, Validation of shell theory for modeling the radial breathing mode of a single-walled carbon nanotube, IJE Transactions A 26(4): 447-454.
[11] Fatahi-Vajari A., Imam A., 2016, Axial vibration of single-walled carbon nanotubes using doublet mechanics, Indian Journal of Physics 90(4): 447-455.
[12] Fatahi-Vajari A., Imam A., 2016, Lateral vibration of single-layered graphine sheets using doublet mechanics, Journal of Solid Mechanics 8(4): 875-894.
[13] Ghorbanpour Arani A., 2015, Surface effect on vibration of Y-SWCNTs embedded on Pasternak foundation conveying viscose fluid, Journal of Nanostructures 5(1): 33-40.
[14] Ghorbanpour Arani A.H., Rastgoo A., Ghorbanpour Arani A., Zarei M. Sh., 2016, Nonlocal vibration of Y-SWCNT conveying fluid considering a general nonlocal elastic medium, Journal of Solid Mechanics 8(2): 232-246.
[15] Ansari R., Gholami R., Rouhi H., 2012, Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories, Composites: Part B 43: 2985-2989.
[16] Ghorbanpour Arani A., Kolahchi R., Jamali M., Mosayyebi M., Alinaghian I., 2017, Dynamic instability of visco-SWCNTs conveying pulsating fluid based on sinusoidal surface couple stress theory, Journal of Solid Mechanics 9(2): 225-238.
[17] Clerck J. D., 2014, Topics in Modal Analysis I , Springer, Cham.
[18] Gopalakrishnan S., Narendar S., 2013, Wave Propagation in Nanostructures, Nonlocal Continuum Mechanics Formulations, Springer, Cham.
[19] Huang J., Han Q., 2016, Controllable nanoscale rotating actuator system based on carbon nanotube and graphene, Nanotechnology 27: 1-9.
[20] Mirtalaie S. H., Hajabasi M. A., 2017, Nonlinear axial-lateral-torsional free vibrations analysis of Rayleigh rotating shaft, Archive of Applied Mechanics 87: 1465-1494.
[21] Cai K., Li Y., Qin Q.H., Yin H., 2014, Gradient less temperature-driven rotating motor from a double-walled carbon nanotube, Nanotechnology 25: 1-6.
[22] Ebrahimi F., Shaghaghi Gh. R., 2015, Vibration analysis of an initially pre-stressed rotating carbon nanotube employing differential transform method, International Journal Advanced Design and Manufacturing Technology 8(4): 13-21.
[23] Ishida Y., Yamamoto T., 2012, Linear and Nonlinear Rotordynamics, Wiley-VCH, Weinheim.
[24] Nahvi H., Boroojeni M.E., 2013, Free Vibrations of a rotating single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Acta Physica Polonica A 123: 304-306.
[25] Hayat T., Haider F., Muhammad T., Alsaedi A., 2017, Three-dimensional rotating flow of carbon nanotubes with Darcy, Forchheimer Porous Medium 12(7):e0179576.1-e0179576.18.
[26] Pradhan S.C., Murmu T., 2010, Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever, Physica E: Low-dimensional Systems and Nanostructures 42(7): 1944-1949.
[27] Murmu T., Adhikari S., 2012, Scale-dependent vibration analysis of prestressed carbon nanotubes undergoing rotation, Journal of Applied Physics 108(12): 123507.1-123507.7.
[28] Narendar S., Gopalakrishnan S., 2011, Nonlocal wave propagation in rotating nanotube, Results in Physics 1(1): 17-25.