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
Related Articles
- EP ID EP462475
- DOI -
- Views 58
- Downloads 0
Dynamics of spacecraft in design deployment of elastic structure of large dimensions
The paper deals with the dynamics of the spacecraft, which carries an elastic variable-geometry body, dur-ing the deployment of a compact system of the circular antenna type in accordance with a given program. A research...
System analysis of space industry projects and orbital complex dynamics and control
This paper analyzes the scientific results obtained at the Department of System Analysis and Control Problems of the Institute of Technical Mechanics of the National Academy of Sciences of Ukraine and the State Space Age...
Synthesis of robust controller for ion beam shepherd control system
The study objective is to synthesize a motion controller for an ion-beam shepherd with respect to space de-bris object during its contactless de-orbiting. It is assumed that the control system has sensors for measuring t...
Development of model of plant for measuring dispersivity of material in energy-carrier flow based on regression analysis
Based on experimental investigations, the plant for measuring the material particle size in the flow has been created. The work objective is to develop a regression model for determining the material dispersivity in the...
Mathematical model of interactions between railway and track considering distributions of contact forces throughout contact spots.
The work objective was to develop a mathematical model of interactions between the railway vehicle and the track considering the distribution of contact forces of the wheel-rail pair throughout contact spots. The meth...