On the destabilizing effect of damping on discrete and continuous circulatory systems

The ‘Ziegler paradox’, concerning the destabilizing effect of damping on elastic systems loaded by nonconservative positional forces, is addressed. The paper aims to look at the phenomenon in a new perspective, according to which no surprising discontinuities in the critical load exist between undamped and damped systems. To show that the actual critical load is found as an (infinitesimal) perturbation of one of the infinitely many sub-critically loaded undamped systems. A series expansion of the damped eigenvalues around the distinct purely imaginary undamped eigenvalues is performed, with the load kept as a fixed, although unknown, parameter. The first sensitivity of the eigenvalues, which is found to be real, is zeroed, so that an implicit expression for the critical load multiplier is found, which only depends on the ‘shape’ of damping, being independent of its magnitude. An interpretation is given of the destabilization paradox, by referring to the concept of ‘modal damping’, according to which the sign of the projection of the damping force on the eigenvector of the dual basis, and not on the eigenvector itself, is the true responsible for stability. The whole procedure is explained in detail for discrete systems, and successively extended to continuous systems. Two sample structures are studied for illustrative purposes: the classical reverse double-pendulum under a follower force and a linear visco-elastic beam under a follower force and a dead load.