Modeling Hysteretically Nonlinear Piezoelectric Composite Beams

Abstract

This paper presents a modeling framework for hysteretically nonlinear piezoelectric composite beams using functional differential equations (FDEs). While linear piezoelectric models are well established, they fail to capture the complex nonlinear behaviors that emerge at higher electric field strengths, particularly history-dependent hysteresis effects. This paper develops a cascade model that integrates a high-dimensional linear piezoelectric composite beam representation with a nonlinear Krasnosel’skii–Pokrovskii (KP) hysteresis operator. The resulting system is formulated using a state-space model where the input voltage undergoes a history-dependent transformation. Through modal expansion and discretization of the Preisach plane, we derive a tractable numerical implementation that preserves essential nonlinear phenomena. Numerical investigations demonstrate how system parameters, including the input voltage amplitude, and hysteresis parameters significantly influence the dynamic response, particularly the shape and amplitude of limit cycles. The results reveal that while the model accurately captures memory-dependent nonlinearities, it depends on numerous real and distributed parameters, highlighting the need for efficient reduced-order modeling approaches. This work provides a foundation for understanding and predicting the complex behavior of piezoelectric systems with hysteresis, with potential applications in vibration control, energy harvesting, and precision actuation.