Mode analysis of trilaminar bender bar transducers using an approximation method

Based on the vibration theory of a thin plate, an analytical treatment of the trilaminar bender bar with piezoelectric elements and inert substrate of various lengths is presented for mode analysis. Resonance frequency and effective electromechanical coupling coefficient are calculated by this method. The impacts of the geometries of the bender bar on the performance of its fundamental and third-order flexural mode are investigated in detail under rigid boundary conditions. It is shown that resonance frequency is extremely sensitive to the thickness of inert substrate. Moreover, the effective electromechanical coupling coefficient has peaks as the length of piezoelectric elements varies. The peaks are achieved when the length of piezoelectric elements equals the length between two nodes having zero strains in the x-direction. The trilaminar bender bar will be effectively excited when the strains on the piezoelectric element are in the same phase, which is important to disclose the vibration mechanisms of this kind of transducer. Also, analytical results are compared with the ones of numerical simulation. The results suggest that effective electromechanical coupling coefficient shares similar patterns with electrical conductance, which can be used to characterize transducer performance to a certain extent. It also demonstrates that the analytical treatment provides an efficient alternative way for optimizing the bender bar transducer design.