On the effects of linear damping on the nonlinear Ziegler’s column

In this paper, the effects of linear damping on the post-critical behavior of the Ziegler’s column are discussed. To this end, the well-known double-pendulum, loaded at the free-end by a follower force, firstly introduced by Ziegler, is considered in regime of finite displacements. The multiple scale method is applied to the equations of motion expanded up to the cubic terms, to analyze the nonlinear behavior of a generically damped column, close to the simple-Hopf bifurcation triggered by the follower force. The obtained bifurcation equations are shown to be useful in providing qualitative information about the nonlinear mechanical response of the column in the whole damping plane. Validation of the asymptotic solution, carried out via numerical analyses of the exact equations of motion, points out the effectiveness of the proposed analysis also on the quantitative side.