Linear and nonlinear damping effects on the stability of the Ziegler column

The destabilizing effect of damping on both linear and nonlinear behavior of the Ziegler column is discussed. The paper addresses classical and non-classical aspects related to the ‘Ziegler paradox’. First, the linear problem is illustrated in a new perspective, according to which no discontinuities in the critical load exist between undamped and damped systems. Second, it furnishes a first overview of the mechanical behavior of the system in the post-critical range. The equations of motion for the system are derived via the extended Hamilton’s principle. Then a linear stability analysis is performed via a perturbation approach, in which, however, simple and not double eigenvalues are perturbed, in contrast with a commonly pursued strategy in the literature. According to this idea, a series expansion around the distinct purely imaginary eigenvalues of the undamped and under-critically loaded system is carried out, with the load kept as a fixed, although unknown, parameter. By pursuing the same idea, an algorithm based on the Multiple Scale Method is developed to investigate the post-critical behavior of the system. The role played by the nonlinear damping on the existence of limit-cycles is discussed.