Critical load and non-linear dynamics of Beck׳s column with electromagnetic actuators

This paper deals with the dynamic stability and near-critical vibrations of Beck׳s column with electromagnetic actuators. The governing equations of the actuators are derived, and the continuous dynamics problem of the column is discretised with the multimodal Galerkin procedure. Firstly, the linear analysis of the stability is carried out. An application of the electromagnetic actuators leads to increase in the critical load and considerably affects the initial flutter frequency. Then the Iooss and Joseph method of construction of the bifurcating solution is used to obtain analytical approximations of the near-critical periodic vibrations. It is shown that the electromagnetic effects lead to a supercritical Hopf bifurcation of the equilibrium state of the column into an orbitally stable limit cycle. Moreover, the actuators allow one to decrease significantly the amplitude of the post-critical vibrations. The effect of magnetic damping strongly depends on the position at which the actuators are attached to the column. Theoretical predictions are verified by numerical simulations of the system.