Nonlinear Vibration of a Magneto-Elastic Cantilever Beam With Tip Mass

Abstract

In this work the effect of the application of an alternating magnetic field on the large transverse vibration of a cantilever beam with tip mass is investigated. The governing equation of motion is derived using D’Alembert’s principle, which is reduced to its nondimensional temporal form by using the generalized Galerkin method. The temporal equation of motion of the system contains nonlinearities of geometric and inertial types along with parametric excitation and nonlinear damping terms. Method of multiple scales is used to determine the instability region and frequency response curves of the system. The influences of the damping, tip mass, amplitude of magnetic field strength, permeability, and conductivity of the beam material on the frequency response curves are investigated. These perturbation results are found to be in good agreement with those obtained by numerically solving the temporal equation of motion and experimental results. This work will find extensive applications for controlling vibration in flexible structures using a magnetic field.