Stiffness matrix method for defective phononic beams toward piezoelectric sensors and actuators under bending waves

Abstract

Defective phononic beams (DPBs) allow for the strong spatial localization of elastic waves, providing the necessary energy amplification to achieve high-sensitivity actuation and sensing. This study presents an analytical framework based on stiffness matrices that can accurately and efficiently model the electromechanically coupled bending wave phenomena in piezoelectric DPBs. Conventional transfer-matrix methods (TMMs) often experience numerical instability and ill-conditioning, particularly in multi-cell or high-frequency configurations. The primary contribution of this study is the development of a numerically stable and computationally efficient stiffness matrix method (SMM). The proposed SMM can predict band structures, defect modes, and electromechanical actuation and sensing responses within a unified analytical framework. Comparisons with finite element methods (FEMs) demonstrate that the SMM achieves accuracy comparable to that of FEMs while producing smooth, physically consistent frequency responses without spurious oscillations. Furthermore, the SMM requires only a fraction of the computational time required for FEM analyses. The formulation is fully implemented in MATLAB, enabling reproducible simulations. Condition-number analyses confirm that, unlike TMMs, the SMM remains well-conditioned across the frequency domain. These findings establish the SMM as a robust, scalable, and time-efficient analytical framework to optimize ultrasonic transducers for nondestructive evaluation, structural health monitoring, and prognostics and health management.