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Variational principles are derived for multi-walled carbon nanotubes (CNT) undergoing non-linear vibrations. Two sources of non-linearity are considered in the continuum modelling of CNT with the Euler-Bernoulli beam model describing the dynamics of the CNT. One source is the geometric non-linearity, which may arise as a result of large deflections. The second source is owing to van der Waals forces between the nanotubes, which can be modelled as a non-linear force to improve the accuracy of the physical model. After deriving the applicable variational principle by the semi-inverse method, Hamilton's principle is given. Natural and geometric boundary conditions are derived using the variational formulation of the problem. Several approximate and computational methods of solution, such as Rayleigh-Ritz and finite elements, employ the variational formulation of the problem and therefore these principles are instrumental in obtaining the solutions of vibration problems under complicated boundary conditions.
Date of Publication: Dec. 2009