Energy localization in carbon nanotubes

In this paper, the energy localization phenomena in low-frequency nonlinear oscillations of single-walled carbon nanotubes (SWNTs) are analysed. The SWNTs dynamics is studied in the framework of the Sanders-Koiter shell theory. Simply supported and free boundary conditions are considered. The effect of the aspect ratio on the analytical and numerical values of the localization threshold is investigated in nonlinear field

Eigenfrequencies and vibration modes of carbon nanotubes

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 needle-like 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 single-walled (SWNTs) and multi-walled carbon nanotubes (MWNTs). The present work is concerned with the analysis of low-frequency linear vibrations of SWNTs: two approaches are presented: a fully analytical method based on a simplified theory and a semi-analytical method based on the theory of thin walled shells. The semi-analytical approach (shortly called “numerical approach”) is based on the Sanders-Koiter shell theory and the Rayleigh-Ritz 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 Rayleigh-Ritz method is then applied to obtain numerically approximate natural frequencies and mode shapes. The second approach is based on a reduced version of the Sanders-Koiter 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 fourth-order 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 (free-free, clamped-free) with extreme accuracy. Both methods are suitable for a forthcoming extension to multi-walled nanotubes and nonlinear vibrations

Nonlinear vibrations and energy distribution of carbon nanotubes

The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter elastic shell theory is applied in order to obtain the elastic strain energy and kinetic energy. The carbon nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields. The theory considers geometric nonlinearities due to large amplitude of vibration. The displacement fields are expanded by means of a double series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. The Rayleigh-Ritz method is applied to obtain approximate natural frequencies and mode shapes. Free boundary conditions are considered. In the nonlinear analysis, the three displacement fields are re-expanded by using approximate eigenfunctions. An energy approach based on the Lagrange equations is considered in order to obtain a set of nonlinear ordinary differential equations. The total energy distribution of the shell is studied by considering combinations of different vibration modes. The effect of the conjugate modes participation is analysed

Transformación del sistema legal en la era digital: planteamiento del problema

This paper discusses the prospects for the development of the legal system, the system of legislation and the judicial system of the Russian Federation in the context of the fourth industrial revolution associated primarily with the digitalization of all spheres of social life, all social institutions, including the law, of course. There is a range of problematic issues that will require their resolution in the near future. It is suggested that the process which can be described as "technologization of law" is inevitable. The possibility of revising the key institutions of law and actually forming new terminology is not ruled out.Este documento analiza las perspectivas para el desarrollo del sistema legal, el sistema legislativo y el sistema judicial de la Federación de Rusia en el contexto de la cuarta revolución industrial asociada principalmente con la digitalización de todas las esferas de la vida social, todas las instituciones sociales, incluidas la ley, por supuesto. Hay una variedad de problemas problemáticos que requerirán su resolución en un futuro cercano. Se sugiere que el proceso que puede describirse como "tecnologización de la ley" es inevitable. No se descarta la posibilidad de revisar las instituciones jurídicas clave y, de hecho, formar una nueva terminología

Digital receivers for low-frequency radio telescopes UTR-2, URAN, GURT

This paper describes digital radio astronomical receivers used for decameter and meter wavelength observations. This paper describes digital radio astronomical receivers used for decameter and meter wavelength observations. Since 1998, digital receivers performing on-the-fly dynamic spectrum calculations or waveform data recording without data loss have been used at the UTR-2 radio telescope, the URAN VLBI system, and the GURT new generation radio telescope. Here we detail these receivers developed for operation in the strong interference environment that prevails in the decameter wavelength range. Data collected with these receivers allowed us to discover numerous radio astronomical objects and phenomena at low frequencies, a summary of which is also presented.Comment: 24 pages, 15 figure

Low-frequency linear vibrations of single-walled carbon nanotubes: Analytical and numerical models

Low-frequency vibrations of single-walled carbon nanotubes with various boundary conditions are considered in the framework of the Sanders-Koiter thin shell theory. Two methods of analysis are proposed. The first approach is based on the Rayleigh-Ritz method, a double series expansion in terms of Chebyshev polynomials and harmonic functions is considered for the displacement fields; free and clamped edges are analysed. This approach is partially numerical. The second approach is based on the same thin shell theory, but the goal is to obtain an analytical solution useful for future developments in nonlinear fields; the Sanders-Koiter equations are strongly simplified neglecting in-plane circumferential normal strains and tangential shear strains. The model is fully validated by means of comparisons with experiments, molecular dynamics data and finite element analyses obtained from the literature. Several types of nanotubes are considered in detail by varying aspect ratio, chirality and boundary conditions. The analyses are carried out for a wide range of frequency spectrum. The strength and weakness of the proposed approaches are shown; in particular, the model shows great accuracy even though it requires minimal computational effort

Transformación del sistema legal en la era digital: planteamiento del problema

This paper discusses the prospects for the development of the legal system, the system of legislation and the judicial system of the Russian Federation in the context of the fourth industrial revolution associated primarily with the digitalization of all spheres of social life, all social institutions, including the law, of course. There is a range of problematic issues that will require their resolution in the near future. It is suggested that the process which can be described as "technologization of law" is inevitable. The possibility of revising the key institutions of law and actually forming new terminology is not ruled out.Este documento analiza las perspectivas para el desarrollo del sistema legal, el sistema legislativo y el sistema judicial de la Federación de Rusia en el contexto de la cuarta revolución industrial asociada principalmente con la digitalización de todas las esferas de la vida social, todas las instituciones sociales, incluidas la ley, por supuesto. Hay una variedad de problemas problemáticos que requerirán su resolución en un futuro cercano. Se sugiere que el proceso que puede describirse como "tecnologización de la ley" es inevitable. No se descarta la posibilidad de revisar las instituciones jurídicas clave y, de hecho, formar una nueva terminología

Legal System Transformation in the Digital Age: Problem Statement

This paper discusses the prospects for the development of the legal system, the system of legislation and the judicial system of the Russian Federation in the context of the fourth industrial revolution associated primarily with the digitalization of all spheres of social life, all social institutions, including the law, of course. There is a range of problematic issues that will require their resolution in the near future. It is suggested that the process which can be described as "technologization of law" is inevitable. The possibility of revising the key institutions of law and actually forming new terminology is not ruled out