Magnus method for dynamic behavior of embedded single-walled carbon nanotube under harmonic excitation

Based on the continuum mechanics and elastic beam model, the nonlinear vibration of embedded single-walled carbon nanotube with clamped-clamped boundary condition is investigated, where the carbon nanotube is modeled as a harmonically excited beam under a transverse force. By using a single-mode Galerkin approximation method, the nonlinear integral-differential equation governing the motion of the nanotube is converted into a second-order nonlinear ordinary differential equation. The differential equation of the model is solved using Magnus expansion method which is one of the geometric integration methods. In the numerical calculation, the amplitude-frequency response curves for embedded single-walled carbon nanotube are analyzed, the effects of the surrounding elastic medium on the amplitude-frequency response characteristics are discussed, the periodic orbits and their bifurcations are obtained, the nonlinear dynamic theory shows that a sequence of period-doubling bifurcations leading to chaos.