A continuum electro-mechanical model of ultrasonic Langevin transducers to study its frequency response

Abstract

In this paper, a continuum mathematical electro-mechanical model of ultrasonic Langevin transducers has been developed by considering three factors: 3D vibrations, piezoelectric effects, and elements damping. Since the equivalent-circuit methods simulate a continuous system with discrete components, they are valid near one resonance frequency. However, a continuum model is valid in broader frequency ranges. It can also describe the effect of geometries and material properties on the frequency response of transducers. By using the mathematical model, the amplitude of vibrations, mode shape, resonant and anti-resonant frequencies, mechanical quality factor, and frequency response can be achieved for a transducer with a specific geometry and material properties. It would also be reliable to design a transducer and to study its frequency response. A transducer is modeled and then numerically simulated with FEM software of COMSOL, and the results are compared with the model. Then the transducer is fabricated and tested experimentally. The results are compared with the mathematical and numerical models. The analytical resonant frequency obtained with the model is 19,378 Hz, which in comparison with 19,270 Hz of the numerical model and 19,457 Hz of the experimental test, has 0.4% and 0.5% errors, respectively. The vibration amplitude of the transducer's tip at the resonant frequency, obtained by the mathematical model, is 21.2 µm, which is 3.6% lower than the 22 µm obtained using experimental tests. The anti-resonant frequency calculated by the mathematical model is 21,748 Hz, which has a 0.7% variation with the experimental result.