Parametric excitation of a piezoelectrically actuated system near Hopf bifurcation

This paper deals with investigation into the stability analysis for transverse motions of a cantilever micro-beam, which is axially loaded due to a voltage applied to the piezoelectric layers located on the lower and upper surfaces of the micro-beam. The piezoelectric layers are pinned to the open end of the micro-beam and not bonded to it through its length. Application of the DC and AC piezoelectric actuations creates steady and time varying axial forces. The equation of the motion is derived using variational principal, and discretized using modal expansion theorem. The differential equations of the discretized model are a set of Mathieu type ODEs, whose stability analysis is performed using Floquet theory for multiple degree of freedom systems. Considering first two eigen-functions in the modal expansion theorem leads in the prediction of flutter type of instability as a consequence of Hopf bifurcation, which is not seen in the reduced single degree of freedom system. The object of the present study is to passively control the flutter instability in the proposed model by applying AC voltage with suitable amplitude and frequency to the piezoelectric layers. The effect of various parameters on the stability of the structure, including damping coefficient, amplitude of the DC and AC voltages, and the frequency of the applied AC voltage is studied.