Linear stability of piezoelectric-controlled discrete mechanical systems under nonconservative positional forces

Discrete models of piezoelectric controlled mechanical systems, subjected to follower forces, are derived in this paper. A targeted strategy of control is first discussed, leading to detect three novel linear controllers resembling, in the order, the Tuned Mass Damper with large and small mass, respectively, and Energy Sink, used in the literature for controlling linear and nonlinear oscillations. Here, however, and differently from those systems, coupling between the controller and the main structure is of gyroscopic type, whose magnitude is governed by an electro-mechanical parameter. Based on an eigenvalue sensitivity analysis, carried out via suitably selected perturbation methods, the effectiveness of the three controllers is investigated. As an application, the two-degree-of-freedom Ziegler column, undergoing Hopf bifurcation triggered by a follower force, equipped with piezoelectric controllers, is studied, and the different strategies proposed numerically compared.