Improving the linear stability of the visco-elastic Beck's beam via piezoelectric controllers

Control strategies for the visco-elastic Beck's beam, equipped with distributed piezoelectric devices and suffering from Hopf bifurcation triggered by a follower force, are proposed in this paper. The equations of motion of the Piezo-Electro-Mechanical (PEM) system are derived through the Extended Hamilton Principle, under the assumption that the piezoelectric patches are shunted to the so-called zero-order network and zero-order analog electrical circuit. An exact solution for the eigenvalue problem is worked out for the PEM system, while an asymptotic analysis is carried out to define three control strategies, recently developed for discrete PEM systems, that are here adapted to improve the linear stability of the visco-elastic Beck's beam. An extensive parametric study on the piezo-electrical quantities, based on an exact linear stability analysis of the PEM system, is then performed to investigate the effectiveness of the controllers.