On the failure of the 'Similar Piezoelectric Control' in preventing loss of stability by nonconservative positional forces

A control strategy for continuous, autonomous, linear mechanical systems, controlled via piezoelectric devices, and suffering Hopf bifurcations, triggered by positional nonconservative forces, is discussed. The strategy is based on the 'principle of similarity', proved in the literature to be successful in controlling externally excited systems. A continuous metamodel of Piezo-Electro-Mechanical system, loaded by position-dependent forces, is derived via the Extended Hamilton Principle. The similarity principle is introduced in the model, demanding for certain relations among mechanical and piezoelectric properties to be satisfied. A stability analysis is carried out via perturbation methods, also accounting for small deviations from similarity. It is shown that the similar control has always a detrimental effect in preventing the loss of stability of the mechanical systems and that this effect is robust under slight imperfections. As an example, a Generalized Beck beam is studied, internally and externally damped, under the simultaneous action of a follower and a dead load, for which the previous asymptotic results are corroborated by an exact analysis.