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.

中文翻译：

---

十字梁面外弯曲振动的二维机电耦合建模


十字型压电梁（CTPB）因其独特的振动形式而广泛应用于换能器、执行器、传感器、能量采集器和振动控制等领域。然而，仍然缺乏描述CTPB二维面外弯曲振动的机电耦合动力学模型。主要原因是现有的数学建模方法无法准确耦合梁两个方向振动的动力学参数。为了解决这个问题，本文提出了一种基于转移矩阵法（TMM）的建模方法。最初，创建了一个新的离散化概念，将 CTPB 的交集离散化为普通弹性梁和无质量弹性梁。随后，建立了离散子单元的弯扭混合振动传递方程。最后，通过推导离散子单元的二维传递条件并结合传递方程，建立了CTPB的机电耦合动力学模型。在本研究中，采用所提出的建模方法建立了横梁奇数阶和偶数阶面外弯曲振动的机电耦合动力学模型作为案例研究。并通过有限元方法和实验研究验证了模型的有效性。此外，对比结果表明，所建立的模型比有限元方法能够更准确地预测振子的阻抗特性。所提出的建模方法填补了半解析方法分析CTPB二维面外振动特性的空白。