Multi-mode piezoelectric shunt damping with residual mode correction by evaluation of modal charge and voltage

A resonant, piezoelectric shunt tuning procedure is based on the limiting eigenvalue problems associated with short and open circuiting of the piezoelectric domains attached to a vibrating structure. Whereas short circuit and open circuit frequencies are directly obtained from the eigenvalues, the associated electrical equation further determines a modal charge as an short circuit reaction force and a new modal voltage from the electric deflection in the open circuit limit. By a modal representation with short circuit mode shapes the structural equation fully decomposes, while the inherent contributions from non-resonant vibration modes in the corresponding electrical equation consistently define an effective modal capacitance, conveniently estimated by the modal charge to voltage ratio. The shunt tuning is obtained from the governing characteristic equation for the targeted vibration mode, in which the residual mode correction is explicitly represented by the effective modal capacitance. The tuning procedure with an effective modal capacitance is generalized to multiple piezoelectric domains with independent shunts for simultaneous damping of multiple vibration modes. The accuracy of the proposed shunt tuning methods is demonstrated by a numerical beam example with three piezoceramic patch pairs and independent resistive–inductive shunts. The analysis is carried out in a commercial finite element program, in which the required modal frequencies, charge and voltage are readily available as output to the short circuit and open circuit eigenvalue problems.