Stabilization of the parametric resonance of a cantilever beam by bifurcation control with a piezoelectric actuator

In this work, bifurcation control using a piezoelectric actuator is implemented to stabilize the parametric resonance induced in a cantilever beam. The piezoelectric actuator is attached to the surface of the beam to produce a bending moment in the beam. The dimensionless equation of motion for the beam with the piezoelectric actuator on its surface is derived and the modulation equations for the complex amplitude of an approximate solution are obtained using the method of multiple scales. We then acquire the bifurcation set that expresses the boundary of the stable and unstable regions. The bifurcation set is characterized by the modulation equations. Next, we determine the order of feedback gains to modify these modulation equations. By actuating the piezoelectric actuator under the appropriate feedback, bifurcation control is carried out resulting in the shift of the bifurcation set and the expansion of the stable region. The main characteristic of the stabilization method introduced above is that the work done by the piezoelectric actuator is zero in the state where the parametric resonance is stabilized. Thus zero power control is realized in such a state. Experimental results show the validity of the proposed stabilization method for the parametric resonance induced in the cantilever beam.