Dynamics of a Flexible Beam With Active Nonlinear Magnetic Force

Abstract

A simply supported beam is controlled in a contactless manner by the attractive force of magnetic actuators that are controlled by feedback signals from displacement and velocity sensors to stabilize the levitation. The equations of motion for the single mode of a beam’s model are used to show that due to the realistic nonlinear terms existing in the magnetic force, the resulting third-order system exhibits codimension-two bifurcations in which a limit cycle and a heteroclinic orbit are created. Using the bifurcation analysis, it was determined how feedback gains influenced the behavior of transverse vibrations of the beam and the feedback gains producing stability were found. The largest region of attraction of those stable regions is showed to exist at certain feedback gain combination. The analytical approach based on the single mode simplification system was proven to be feasible via the numerical simulation of the governing nonlinear PDE. According to the results of the analysis, due to the nonlinear magnetic force, feedback gains for the system must be selected carefully.