Modeling of coupled longitudinal and bending vibrations in a sandwich type piezoelectric transducer utilizing the transfer matrix method

Abstract Sandwich type piezoelectric transducers are widely employed as actuating mechanisms for ultrasonic motors and ultrasonic cutting machines due to their advantages of compact structure, no electromagnetic interference, excellent mechanical performance, and fast response. By simultaneously adopting bending piezoelectric ceramics and longitudinal piezoelectric ceramics, sandwich type piezoelectric transducers can be utilized to generate coupled longitudinal and bending vibrations. To neglect the specific and complex polarization of the bending piezoelectric ceramics, a novel sandwich type piezoelectric transducer adopting commonly rectangular longitudinal piezoelectric ceramics is proposed in this study. The proposed transducer can be stimulated to produce the coupled longitudinal and bending vibrations by applying two electrical signals with shifted phase. To reveal the dynamic behavior of the proposed transducer and reduce the computational efforts of the finite element simulation, a semi-analytical model is developed using the transfer matrix method. Although the individual longitudinal or bending vibration model has been developed for piezoelectric elements, the modeling of coupled longitudinal and bending vibrations by simultaneously considering the electrical and mechanical coefficients is still unavailable. Therefore, a new transfer matrix model is created for the composite piezoelectric beam to describe the coupled longitudinal and bending vibrations. The presented transfer matrix model is capable of optimizing the proposed transducer and is also suitable for modeling conventional sandwich type piezoelectric transducers. To validate the effectiveness of the proposed model, two case studies are conducted. First, the optimizations of the transducer are conducted to obtain suitable geometrical dimensions. Then the frequency response characteristics and vibration shapes of the transducer are computed and compared to finite element simulation results. The comparisons demonstrate that the proposed transfer matrix model is valid and can effectively reduce the computational efforts.