Asymptotic Analysis of Nonlinear Modes of Multi-Walled Nanotubes

Sanders-Koiter nonlinear shell theory is applied to obtain the partial differential equations, which describe the nanotube nonlinear oscillations. Three groups of nonlinearities are taken into account: the geometrical nonlinearity, Van der Waals nonlinearity and boundary condition nonlinearity. Using the weight residual method, the partial differential equations are transformed into the nonlinear ordinary differential equations, which have several internal resonances. The multiple-mode manifolds are applied to study nonlinear oscillations. Due to these manifolds, the finite degree of freedom nonlinear mechanical system is reduced into two degree of freedom system, which is analyzed by the asymptotic approach. The data of numerical analysis is presented on backbone curves.