Unified nonlinear electroelastic dynamics of a bimorph piezoelectric cantilever for energy harvesting, sensing, and actuation

Inherent nonlinearities of piezoelectric materials are pronounced in various engineering applications such as sensing, actuation, combined applications for vibration control, and energy harvesting from dynamical systems. The existing literature focusing on the dynamics of electroelastic structures made of piezoelectric materials has explored such nonlinearities separately for the problems of mechanical and electrical excitation. Similar manifestations of softening nonlinearities have been attributed to purely elastic nonlinear terms, coupling nonlinearities, hysteresis alone, or a combination of these effects by various authors. In order to develop a unified nonlinear nonconservative framework with two-way coupling, the present work investigates the nonlinear dynamic behavior of a bimorph piezoelectric cantilever under low to moderately high mechanical and electrical excitation levels in energy harvesting, sensing, and actuation. The highest voltage levels, for near resonance investigation, are well below the coercive field. A distributed parameter electroelastic model is developed by accounting for softening and dissipative nonlinearities to analyze the primary resonance of a soft (e.g., PZT-5A, PZT-5H) piezoelectric cantilever for the fundamental bending mode using the method of harmonic balance. Excellent agreement between the model and experimental investigation is found, providing evidence that quadratic stiffness, damping, and electromechanical coupling effects accurately model predominantly observed nonlinear effects in geometrically linear vibration of piezoelectric cantilever beams. The backbone curves of both energy harvesting and actuation frequency responses for a PZT-5A cantilever are experimentally found to be dominantly of first order and specifically governed by ferroelastic quadratic softening for a broad range of mechanical and electrical excitation levels. Cubic and higher-order nonlinearities become effective only near the physical limits of the brittle and stiff (geometrically linear) bimorph cantilever when excited near resonance.