Extension of the fixed point theory to tuned mass dampers with piezoelectric stack energy harvester

A tuned mass damper with a piezoelectric stack energy harvester (TMD-PSEH) is a passive vibration control device for the dual purpose of absorbing vibration and harvesting energy. This paper develops the fixed point theory for the optimal design of a TMD-PSEH attached to an undamped primary structure under harmonic external excitation. It has been proven that the existence of two fixed points of the amplitude-frequency curve doest not depend on the damping of the primary structure. Based on the requirements for suppressing the vibration of the primary structure and enlarging the harvested power, the analytical expressions for the optimal tuning, damping, and resistance ratios are determined. It is found that the minimum damping ratio occurs in the vicinity of the electrical resistance ratio alpha = 1 for various values of the electromechanical coupling coefficient. Power flow analysis is carried out resulting in closed-form expressions of instantaneous and average powers. Numerical examinations of the system with the obtained optimal parameters are performed demonstrating a very good compatibility between theory and computation. In addition, the effect of damping in the primary structure is also investigated. It is shown that not only the vibration of the primary structure effectively reduces, but also a large amount of the harvested energy can be captured in the main resonance region.