Vibration control of a cantilever beam subject to both external and parametric excitation

The problem of suppressing the vibrations of a structure subjected to parametric and external excitation is studied at resonance. The vibration amplitudes resulting from such resonance cannot be fully controlled by conventional techniques, such as the increase of 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 such non-linearities. In this work, this fact is obtained on a simple non-linear feedback law, which is devised to suppress the vibration of the first mode of a cantilever beam subjected to a primary and sub-harmonic resonance. The dynamics of the beam are modeled by a second order non-linear ordinary differential equation. A control law based on cubic velocity feedback is proposed. The method of multiple scales is used to drive two first order ordinary differential equations, which are used to obtain the time variation of both the amplitude and the phase at resonance. The stability and effects of different parameters are studied numerically. (C) 2003 Elsevier Inc. All rights reserved.