Forced Response Analysis of a Piezoelectric Flag Modeled as a Membrane

A piezoelectric flag is modeled as a membrane for energy harvesting purposes. The tension, pressure, piezoelectric material, and external force are introduced and employed in the model. In this model, the tension is caused by gravity, the piezoelectric material, and the fluid flow. The pressure acting on the flag consists of a non-circulatory and a circulatory component. Additionally, an external force is modeled to ensure that the pressure acting on the system does not dissipate. To model the system, Hamilton’s principle is employed to find differential equation of motion. In this study, the flag is vertically oriented. This is to ensure the flag does not droop, which would greatly complicate the effect of gravity. In studying the free response, it is found that the Bessel function of the first kind describes the flag. Lastly, Galerkin’s method is applied to the system. This allows for the deflection and the voltage produced by the flag to be found. It is found that the presented model reasonably predicts both the deflection and voltage of the piezoelectric flag.