In 1991 Iijima discovered Carbon Nanotubes, he synthesised molecular carbon structures in the form of fullerenes and then reported the preparation of a new type of finite carbon structure consisting of needlelike tubes, the carbon nanotubes, described as helical microtubules of graphitic carbon. Examples of applications of Carbon Nanotubes (CNTs) can be found in ultrahigh frequency nanomechanical resonators, in a large number of nanoelectromechanical devices such as sensors, oscillators, charge detectors and field emission devices. The reduction of the size and the increment of the stiffness of a resonator magnify its resonant frequencies and reduce its energy consumption, improving its sensitivity. The modal analysis of carbon nanotubes is important because it allows to obtain the resonant frequencies and mode shapes, which influence the mechanical and electronic properties of the nanotube resonators. A large number of experiments and atomistic simulations were conducted both on singlewalled (SWNTs) and multiwalled carbon nanotubes (MWNTs). The present work is concerned with the analysis of lowfrequency linear vibrations of SWNTs: two approaches are presented: a fully analytical method based on a simplified theory and a semianalytical method based on the theory of thin walled shells. The semianalytical approach (shortly called “numerical approach”) is based on the SandersKoiter shell theory and the RayleighRitz numerical procedure. The nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields, which are expanded by means of a double mixed series based on Chebyshev polynomials for the longitudinal variable and harmonic functions for the circumferential variable. The RayleighRitz method is then applied to obtain numerically approximate natural frequencies and mode shapes. The second approach is based on a reduced version of the SandersKoiter shell theory, obtained by assuming small ring and tangential shear deformations. These assumptions allow to condense both the longitudinal and the circumferential displacement fields. A fourthorder partial differential equation for the radial displacement field is derived. Eigenfunctions are formally obtained analytically, then the numerical solution of the dispersion equation gives the natural frequencies and the corresponding normal modes. The methods are fully validated by comparing the natural frequencies of the SWNTs with data available in literature, namely: experiments, molecular dynamics simulations and finite element analyses. A comparison between the results of the numerical and analytical approach is carried out in order to check the accuracy of the last one. It is worthwhile to stress that the analytical model allows to obtain results with very low computational effort. On the other hand the numerical approach is able to handle the most realistic boundary conditions of SWNTs (freefree, clampedfree) with extreme accuracy. Both methods are suitable for a forthcoming extension to multiwalled nanotubes and nonlinear vibrations.
Eigenfrequencies and vibration modes of carbon nanotubes / Strozzi, Matteo; Manevitch, Leonid I.; Smirnov, Valeri V.; Shepelev, Denis S.; Pellicano, Francesco.  (2014). ((Intervento presentato al convegno 12th International Conference on Computational Structures Technology CST 2014 tenutosi a Naples, Italy nel September 0205, 2014.
Eigenfrequencies and vibration modes of carbon nanotubes
STROZZI, MATTEO;PELLICANO, Francesco
2014
Abstract
In 1991 Iijima discovered Carbon Nanotubes, he synthesised molecular carbon structures in the form of fullerenes and then reported the preparation of a new type of finite carbon structure consisting of needlelike tubes, the carbon nanotubes, described as helical microtubules of graphitic carbon. Examples of applications of Carbon Nanotubes (CNTs) can be found in ultrahigh frequency nanomechanical resonators, in a large number of nanoelectromechanical devices such as sensors, oscillators, charge detectors and field emission devices. The reduction of the size and the increment of the stiffness of a resonator magnify its resonant frequencies and reduce its energy consumption, improving its sensitivity. The modal analysis of carbon nanotubes is important because it allows to obtain the resonant frequencies and mode shapes, which influence the mechanical and electronic properties of the nanotube resonators. A large number of experiments and atomistic simulations were conducted both on singlewalled (SWNTs) and multiwalled carbon nanotubes (MWNTs). The present work is concerned with the analysis of lowfrequency linear vibrations of SWNTs: two approaches are presented: a fully analytical method based on a simplified theory and a semianalytical method based on the theory of thin walled shells. The semianalytical approach (shortly called “numerical approach”) is based on the SandersKoiter shell theory and the RayleighRitz numerical procedure. The nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields, which are expanded by means of a double mixed series based on Chebyshev polynomials for the longitudinal variable and harmonic functions for the circumferential variable. The RayleighRitz method is then applied to obtain numerically approximate natural frequencies and mode shapes. The second approach is based on a reduced version of the SandersKoiter shell theory, obtained by assuming small ring and tangential shear deformations. These assumptions allow to condense both the longitudinal and the circumferential displacement fields. A fourthorder partial differential equation for the radial displacement field is derived. Eigenfunctions are formally obtained analytically, then the numerical solution of the dispersion equation gives the natural frequencies and the corresponding normal modes. The methods are fully validated by comparing the natural frequencies of the SWNTs with data available in literature, namely: experiments, molecular dynamics simulations and finite element analyses. A comparison between the results of the numerical and analytical approach is carried out in order to check the accuracy of the last one. It is worthwhile to stress that the analytical model allows to obtain results with very low computational effort. On the other hand the numerical approach is able to handle the most realistic boundary conditions of SWNTs (freefree, clampedfree) with extreme accuracy. Both methods are suitable for a forthcoming extension to multiwalled nanotubes and nonlinear vibrations.File  Dimensione  Formato  

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