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

Abstract

Inherent nonlinearities of piezoelectric materials are inevitably pronounced in various engineering applications such as sensing, actuation, their combined applications for vibration control, and most recently, energy harvesting from dynamical systems. The existing literature focusing on the dynamics of electroelastic structures made of piezoelectric materials have explored such nonlinearities in a disconnected way for the separate problems of mechanical and electrical excitation such that nonlinear resonance trends have been assumed to be due to different additional terms in constitutive equations by different researchers. Similar manifestations of softening nonlinearities have been attributed to purely elastic nonlinear terms, coupling nonlinearities, hysteresis, or a combination of these effects, by various authors. However, a reliable nonlinear constitutive equation for a given piezoelectric material is expected to be rather unique and valid regardless of the application, e.g. energy harvesting, sensing, or actuation. A systematic approach focusing on the two-way coupling can result in a sound mathematical framework. To this end, the present work investigates the nonlinear dynamic behavior of a bimorph piezoelectric cantilever under low-to-high mechanical and electrical excitation levels in energy harvesting, sensing, and actuation. A physical model is proposed including both ferroelastic hysteresis, stiffness, and electromechanical coupling nonlinearities. A lumped parameter electroelastic model is developed by accounting for these nonlinearities to analyze the primary resonance of a cantilever using the method of harmonic balance. Strong agreement between the model and experimental investigation is found, providing solid evidence that the the dominant source of observed softening nonlinear effects in geometrically linear piezolectric cantilever beams is well represented by a quadratic term resulting from ferroelastic hysteresis. Electromechanical coupling and cubic softening nonlinearities are observed to become effective only near the physical limits of the brittle and stiff bimorph cantilever used in the experiments, revealing that the quadratic nonlinearity associated with hysteresis has the primary role in nonlinear nonconservative dynamic behavior.