Nonlinear dynamics and stability analysis of a parametrically excited CNT-reinforced MRE viscoelastic cantilever beam

Abstract

This paper investigates the dynamic response of a clamped-free CNT-reinforced-MRE beam which is actuated by the combination of a constant and a harmonic time-dependent magnetic field. Using Hamilton's principle, the equation of motion has been obtained and discretized using the Galerkin method. This procedure transforms the governing PDE equation of motion into a nonlinear ODE equation in the form of the nonlinear Mathieu equation with cubic damping. Then, the method of multiple scales is employed to obtain the dynamic response of the system. Furthermore, a stability analysis is also performed and the effects of a magnetic field on the dynamic response and stability of the system is investigated. The stability analysis shows that as the amplitude of the constant magnetic field is increased, the stable region decreases. Also, a numerical bifurcation analysis has been performed and Feigenbaum diagrams are obtained, indicating that when the constant magnetic field is less than the harmonic one, the system approaches greater amplitudes and undergoes more chaos, and vice versa.