Study on the longitudinal-radial coupled vibration of cascade cylindrical piezoelectric transducers

Cascade cylindrical piezoelectric transducers (CCPTs) are widely used in ultrasonic engineering, and accurate vibration analysis is essential for understanding their operating mechanisms. Traditional one-dimensional (1D) vibration theory assumes the transducer's length significantly exceeds its diameter, neglecting radial vibrations. However, in practical applications, radial dimensions influence the overall vibration behavior and must be considered. In this paper, the equivalent elastic method is applied to analyze the coupled longitudinal–radial vibration of CCPTs by introducing a mechanical coupling coefficient. An electromechanical equivalent circuit is formulated, from which resonance frequency equations and the effective electromechanical coupling coefficient are derived. Numerical and experimental results demonstrate that the method agrees well with finite element method (FEM) results and predicts the vibration behavior more accurately than the 1D theory. In addition, compared with FEM, the proposed method provides clearer physical insight into the coupled vibration mechanism. The results also show that when the longitudinal and radial dimensions approach each other, the mechanical coupling becomes strong and the radial dimension must be considered in resonance analysis.