Two-dimensional electromechanical-coupling modeling for out-of-plane bending vibration of cross-type beams

Cross-type piezoelectric beams (CTPBs) have been widely employed in the fields of transducers, actuators, sensors, energy harvesters, and vibration control due to their unique vibration form. However, there is still a lack of electromechanical-coupling dynamic model for describing the two-dimensional out-of-plane bending vibration of CTPBs. The main reason is that the current mathematical modeling methods cannot accurately couple the dynamical parameters of beam vibrations in both directions. To solve this problem, a novel modeling method based on the transfer matrix method (TMM) is proposed in this paper. Initially, a new discretization concept is created to discretize the intersection of CTPBs into ordinary and massless elastic beams. Subsequently, the bending-torsional hybrid vibration transfer equations for discrete sub-elements are established. Finally, by deriving the two-dimensional transfer conditions of discrete sub-elements and combining the transfer equations, an electromechanical-coupling dynamic model of CTPBs is developed. In this study, the proposed modeling method is used to establish the electromechanical-coupling dynamic models of odd- and even-order out-of-plane bending vibrations of cross beams as case studies. And the validity of the models is verified through both finite element method and experimental study. Moreover, comparison results indicate that the established models can predict the impedance characteristics of the vibrators more accurately than finite element methods. The proposed modeling method fills the gap of the semi-analytical method in analyzing the two-dimensional out-of-plane vibration characteristics of CTPBs.