A paradigmatic minimal system to explain the Ziegler paradox

The destabilization effect of damping on a class of general dynamical systems is discussed. The phenomenon of jump in the critical value of the bifurcation parameter, in passing from undamped to damped system, is view in a new perspective, according to which no discontinuities manifest themselves. By using asymptotic analysis, it is proved that all subcritically loaded undamped systems are candidate to become unstable, provided a suitable damping matrix is added. The mechanism of instability is explained by introducing the concept of modal dampings, as the components of the damping forces along the unit vectors of a non-orthogonal eigenvector basis. Such quantities can change sign while the load changes the eigenvectors of the basis, thus triggering instability. A paradigmatic, non-physical, minimal system has been built up, admitting closed-form solutions able to explain the essence of the destabilizing phenomenon. Series expansions carried out on the exact solution give information on how to deal more complex systems by perturbation methods.