Vibration Properties and High Speed Stability of a Rotating Piezoelectric Energy Harvesting Device That Experiences Gyroscopic Effects

This study investigates the vibration of a rotating piezoelectric device that consists of a proof mass that is supported by elastic structures with piezoelectric layers. Vibration of the proof mass causes deformation in the piezoelectric structures and voltages to power electrical loads. The coupled electromechanical equations of motion are derived using Newtonian mechanics and Krichhoff\u27s circuit laws. The free vibration behavior is investigated for devices with identical (tuned) and nonidentical (mistuned) piezoelectric support structures and electrical loads. These devices complex-valued have speed-dependent eigenvalues and eigenvectors as a result of their constant rotation. The imaginary parts of the eigenvalues physically represent the oscillation frequencies of the device. The real parts represent the decay or growth rates of the oscillations, depending on their sign. The complex-valued components of the eigenvectors physically represent the amplitudes and phases of the vibration. The eigenvalue behaviors differ for tuned and mistuned devices. Due to gyroscopic effects, the proof mass in the tuned device only vibrates in either forward or backward decaying circular orbits in single-mode free response. This is proven analytically for all tuned devices. For mistuned devices, the proof mass has decaying elliptical forward and backward orbits. The eigenvalues are shown to be sensitive to changes in the electric load resistances. Closed-form solutions for the eigenvalues are derived for open and close circuits. At high rotation speeds these devices experience critical speeds and instability. Closed-form solutions for the critical speeds are derived. Tuned devices have one degenerate critical speed that separates stable speeds from unstable speeds, where flutter instability occurs. Mistuned devices have two critical speeds. The first critical speed separates stable speeds from speeds where divergence instability occurs. Divergence instability continues for small speeds above the second critical speed. At higher supercritical speeds flutter instability occurs. Transitions between stable and unstable eigenvalues are investigated using root locus diagram. This device has atypical eigenvalue behavior near the critical speeds and stability transitions compared to conventional gyroscopic systems