Piezoelectric control of the Hopf bifurcation of Ziegler's column with nonlinear damping

Linear and nonlinear analyses of a piezoelectric-controlled Ziegler's column, endowed with a Van der Pol-like nonlinear damping, are carried out in the present paper. The effects of three linear and passive piezoelectric controllers on the position and on the amplitude of the limit-cycle at the Hopf bifurcation, triggered by the follower force, are investigated. The controllers, previously introduced in the literature and here referred to as Non-Singular Resonant, Singular Non-Resonant, and Tuned Piezoelectric Damper, resemble well-known devices, adopted for controlling both linear and nonlinear oscillations, namely, a large-mass Tuned Mass Damper, a singular Energy Sink, and a classical small-mass Tuned Mass Damper. Numerical simulations on a linearly and uniformly damped column, under different nonlinear damping conditions, are carried out to show the effectiveness of the controllers in reducing the amplitude of the limit-cycle and their behavior with respect to the possible occurrence of the hard loss bifurcation.