Single-Mode Control of a Cantilever Beam Under Principal Parametric Excitation

Abstract

The problem of suppressing the vibrations of a structure that is subjected to a principal parametric excitation is tackled. The vibration amplitudes resulting from such resonance cannot be fully controlled by conventional techniques, such as the addition of linear damping through velocity feedback or by the implementation of conventional mass absorbers. However, it has been shown that the growth of the response is limited by non-linearities. In this work, this fact is capitalized on and a simple non-linear feedback law is devised to suppress the vibrations of the first mode of a cantilever beam when subjected to a principal parametric resonance. The dynamics of the beam are modelled with a second-order non-linear ordinary-differential equation. The model accounts for viscous damping, air drag, and inertia and geometric non-linearities. A control law based on cubic velocity feedback is proposed. The method of multiple scales is used to derive two first-order ordinary-differential equations that govern the time variation of the amplitude and phase of the response. A stability study is conducted and the open- and closed-loop response of the system is analyzed. Furthermore, results are presented of experiments conducted to control the vibrations of a cantilever steel beam fitted with piezoceramic actuators. The theoretical and experimental findings indicate that the control law leads to effective vibration suppression and bifurcation control.