Hard loss of stability of Ziegler’s column with nonlinear damping

The paper is devoted to discuss the effects of nonlinear damping on the post-critical behavior of the Ziegler’s column. The classical Ziegler’s double-pendulum is considered in regime of finite displacements, in which, moreover, nonlinear damping of Van der Pol-type is introduced at the hinges. A second-order Multiple Scale analysis is carried out on the equations of motion expanded up to the fifth-order terms. The nature of the Hopf bifurcation, namely, supercritical or subcritical, as well as the occurrence of the ‘hard loss of stability’ phenomenon, are investigated. The effects of the nonlinear damping on the amplitude of the limit-cycle are finally studied for different linearly damped columns.