Nonlinear Normal Modes of Oscillating Nanotubes

Single walled nanotube, which is described by the geometrical nonlinear Sanders-Koiter shell theory, is considered. The free nonlinear oscillations of the nanoshell are described by the system of nonlinear partial differential equations. These equations are reduced to the nonlinear system of the ordinary differential equations. The concept of nonlinear normal modes is used to study the free nonlinear oscillations. Three approaches (harmonic balance method, Shaw-Pierre concept and shooting technique) are used to analyze nonlinear normal modes of nanotubes. These three approaches are used to analyze the backbone curves of nonlinear normal modes. The Naimark-Sacker bifurcations are observed on this curve. As a result of this bifurcation, the quasi-periodic motions are observed.