Shell continuum model of free nonlinear vibrations of carbon nanotubes with account for nonlocal elasticity
Journal Title: Технічна механіка, Техническая механика, Technical Mechanics - Year 2018, Vol 2, Issue 2
Abstract
In this paper, a model of nonlinear vibrations of a carbon nanotube based on the shell theory is presented. On the basis of variational principles, a system of three nonlinear partial differential equations in three projections of middle surface point displacements was derived. In doing so, use was made of the geometrically nonlinear Sanders–Koiter shell theory and nonlocal elasticity, which modifies the form of Hooke’s law. It was assumed that conjugate vibrational modes are involved in shell vibrations under nonlinear deformation. This assumption together with the Galerkin approach made it possible to derive a nonlinear system of ordinary differential equations in generalized coordinates of the structure, which describes free nonlinear vibrations of the nanotube. The dynamic system obtained contains quadratic and cubic nonlinear terms. To calculate the free nonlinear vibrations, use was made of the harmonic balance method, in which vibrations are represented as a Fourier series. Using this method, the backbone curves of the free nonlinear vibrations were calculated. The backbone curves proved to be soft. The stability of the periodic vibrations obtained was analyzed by direct numerical integration of the motion equations. It was shown that free nonlinear vibrations of carbon nanotubes lose stability due to the Naimark–Saker bifurcation, and almost periodic vibrations are set up due to this bifurcation. These almost periodic motions were studied using Poincare sections. The Poincare section calculations showed that an invariant torus is formed in the system. The calculated almost periodic vibrations are shown in bifurcational diagrams. The longitudinal vibrations and the flexural motions have comparable amplitudes. These properties of nanotubes are new.
Authors and Affiliations
K. Avramov
Keywords
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