Characterization of Hysteretic Response and Multiple Solutions of a Vortex-Induced Vibration Based Piezoelectric Harvester

In this work, a distributed-parameter model describing the flow-structure-electrical interaction in a piezoelectric-based vortex-induced vibration energy harvester is presented. For the closest resemblance with the physical configuration, the model incorporates a compound mode-shape accounting for partial lengths of the piezo-composite and only substrate material sections of the cantilever. Bluff-body mass at the cantilever tip, its rotational inertia, and nonlinear geometry effect are considered for modeling. The Euler-Lagrange equation-based model is simulated using the continuous and discrete numerical simulations with high-and low-energy initial conditions. The results show that the continuous forward and reverse wind velocity sweeps effectively capture the frequency lock-in region along-with the hysteretic multiple solution regions in it. Presence of hysteretic regions with multiple solutions reduced the effective range of lock-in region and introduced a dependency on the initial conditions. The model results are validated using the experimental and numerical results from the literature. Dynamical characterization of the identified multiple solutions is performed by analyzing the time-history, frequency spectrum, and phase-portraits of the responses. Dynamical tools such as the Stroboscopic points and Poincaré maps are used to determine the periodicity of the solutions. Further, influence of the detuning in the forced van der Pol equation on the dynamical characteristics of the solutions during the lock-in state is analyzed. These analyses revealed the relative influence of structural and vortex shedding frequencies on the response frequency and periodicity of the solutions. As the presence of hysteretic regions reduce the effective width of the frequency lock-in region, the results of characterization presented through this investigation are quite essential from the design perspective.