Nonlinear Vibration Analysis of the Fluid-Filled Single Walled Carbon Nanotube with the Shell Model Based on the Nonlocal Elacticity Theory

Document Type : Research Paper


Department of Mechanical Engineering, Semnan branch, Islamic Azad university


Nonlinear vibration of a fluid-filled single walled carbon nanotube (SWCNT) with simply supported ends is investigated in this paper based on Von-Karman’s geometric nonlinearity and the simplified Donnell’s shell theory. The effects of the small scales are considered by using the nonlocal theory and the Galerkin's procedure is used to discretize partial differential equations of the governing into the ordinary differential equations of motion. To achieve an analytical solution, the method of averaging is successfully applied to the nonlinear governing equation of motion. The SWCNT is assumed to be filled by the fluid (water) and the fluid is presumed to be an ideal non compression, non rotation and in viscid type.  The fluid-structure interaction is described by the linear potential flow theory. An analytical formula was obtained for the nonlinear model and the effects of an internal fluid on the coupling vibration of the SWCNT-fluid system with the different aspect ratios and the different nonlinear parameters are discussed in detail. Furthermore, the influence of the different nonlocal parameters on the nonlinear vibration frequencies is investigated according to the nonlocal Eringen’s elasticity theory.


[1] Lee S.M., An K.H., Lee Y.H., Seifert G., Frauenheim T., 2001, A hydrogen storage mechanism in single-walled carbon nanotubes, Journal of the American Chemical Society 123: 5059-5063.
[2] Hummer G., Rasaiah J. C., Noworyta J.P., 2001, Water conduction through the hydrophobic channel of carbon nanotubes, Nature 414: 188-190.
[3] Liu J., Rinzler A. G., Dai H.J., 1998, Fulleren pipes, Science 280: 1253-1256.
[4] Yakobson B.I., Brabec C.J., Bernholc J., 1996, Nanomechanics of carbon tubes: instabilities beyond linear response, Physical Review Letters 76: 2511-2514.
[5] Yoon J., Ru C. Q., Mioduchowski A., 2005, Vibration and instability of carbon nanotubes conveying fluid, Composites Science and Technology 65: 1326-1336.
[6] Yan Y., Huang X.Q., Zhang L.X., Wang Q., 2007, Flow-induced instability of double-walled carbon nanotubes based on an elastic shell model, Journal of Applied Physics 102: 044307.
[7] Yan Y., Wang W.Q., Zhang L.X., He X.Q., 2009, Dynamical behaviors of fluid- conveyed multi-walled carbon nanotubes, Applied Mathematical Modelling 33: 1430-1440.
[8] Wang L., Ni Q., 2008, On vibration and instability of carbon nanotubes conveying fluid, Computational Materials Science 43: 399-402.
[9] khosravian N., Rafii-Tabar H., 2008, Computational modelling of a non-viscous fluid flow in a multi-walled carbon nanotube modelled as a Timoshenko beam, Nanotechnology 19: 275703.
[10] Khadem S.E., Rasekh M., 2009, Nonlinear vibration and stability analysis of axially loaded embedded carbon nanotubes conveying fluid, Applied Physics 42: 135112.
[11] Ghavanloo E., MasoudRafiei F., 2010, Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation, Physica E 42: 2218-2224.
[12] Dong K., Liu B.Y., Wang X., 2008, Wave propagation in fluid-filled multi-walled carbon nanotubes embedded in elastic matrix, Computational Materials Science 42: 139-148.
[13] Yan Y., Wang W.Q., Zhang L.X., 2010, Noncoaxial vibration of fluid-filled multi-walled carbon nanotubes, Applied Mathematical Modelling 34: 122-128.
[14] Yan Y., Wang W.Q., Zhang L.X., 2009, Nonlinear vibration chara cristics of fluid- filled double-walled carbon nanotubes, Modern Physics Letters B 23: 2625-2636.
[15] Eringen A.C., 2002, Nonlocal Continuum Field Theories, New York, Springer.
[16] Eringen A.C., 1976, Nonlocal Polar Field Models, New York, Academic Press.
[17] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 4703-4710.
[18] 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 inextensional modes of vibration, Computational Materials Science 47: 1049-1059.
[19] Amabili M., 2008, Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press.
[20] Karagiozisa K.N., Amabili M., Paı¨doussisa M.P., Misra A.K., 2005, Nonlinear vibrations of fluid-filled clamped circular cylindrical shells, Journal of Fluids and Structures 21: 579-595.
[21] Pellicano F., Amabili M., 2003, Stability and vibration of empty and fluid-filled circular cylindrical shells under static and periodic axial loads, International Journal of Solids and Structures 40: 3229-3251.
[22] Goncalves P.B., Batista R.C, 1998, Non-linear vibration analysis of fluid-filled cylindrical shells, Journal of Sound and Vibration 127: 133-143.
[23] Amabili M., 2005, Non-linear vibrations of doubly curved shallow shells, International Journal of Non-Linear Mechanics 40: 683-710.
[24] Amabili M., Pellicano F., Paidoussis M.P., 1998, Nonlinear vibrations of simply supported circular cylindrical shells, coupled to quiescent fluid, Journal of Fluids and Structures 12: 883-918.
[25] Evensen D.A.,1967, Nonlinear Flexural Vibrations of Thin-Walled Circular Cylinders, National Aeronautics and Space Administration, Spring field.
[26] Nayfeh A.H., Mook D.T., 1995, Nonlinear Oscillations, Wiley.
[27] Liu D.K., 1998, Nonlinear Vibrations of Imperfect Thin-Walled Cylindrical Shells.
[28] Yan Y., Wang W., Zhang L.,2012, Free vibration of the fluid-filled single-walled carbon nanotube based on a double shell-potential flow model, Applied Mathematical Modeling 36: 6146-6153.
[29] Narendar S. , Roy Mahapatra D., Gopalakrishnan S., 2011, Prediction of nonlocal scaling parameter for armchair and zigzag single-walled carbon nanotubes based on molecular structural mechanics, nonlocal elasticity and wave propagation, International Journal of Engineering Science 49: 509-522.