Semi-inverse method in nonlinear mechanics: application to couple shell- and beam-type oscillations of single-walled carbon nanotubes

Abstract

We demonstrate the application of the efficient semi-inverse asymptotic method to resonant interaction of the nonlinear normal modes belonging to different branches of the CNT vibration spectrum. Under condition of the 1:1 resonance of the beam and circumferential flexure modes we obtain the dynamical equations, the solutions of which describe the coupled stationary states. The latter are characterized by the non-uniform distribution of the energy along the circumferential coordinate. The non-stationary solutions for obtained equations correspond to the slow change of the energy distribution. It is shown that adequate description of considered resonance processes can be achieved in the domain variables. They are the linear combinations of the shell- and beam-type normal modes. Using such variables we have analyzed not only nonlinear normal modes but also the limiting phase trajectories describing the strongly non-stationary dynamics. The evolution of the considered resonance processes with the oscillation amplitude growth is analyzed by the phase portrait method and verified by the numerical integration of the respective dynamical equations.