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 19378 Hz, which in comparison with 19270 Hz of the numerical model and 19457 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 microns, which is 3.6% lower than the 22 microns obtained using experimental tests. The anti-resonant frequency calculated by the mathematical model is 21748 Hz, which has a 0.7% variation with the experimental result.

中文翻译：

---

超声朗之万换能器的连续机电模型研究其频率响应

摘要 在本文中，通过考虑三个因素：3D 振动、压电效应和元件阻尼，开发了超声波朗之万换能器的连续数学机电模型。由于等效电路方法模拟具有分立元件的连续系统，因此它们在一个谐振频率附近是有效的。然而，连续模型在更宽的频率范围内是有效的。它还可以描述几何形状和材料特性对换能器频率响应的影响。通过使用数学模型，可以为具有特定几何形状和材料特性的换能器实现振动幅度、模式形状、谐振和反谐振频率、机械品质因数和频率响应。设计换能器并研究其频率响应也是可靠的。对换能器进行建模，然后使用 COMSOL 有限元软件进行数值模拟，并将结果与​​模型进行比较。然后制造换能器并进行实验测试。结果与数学和数值模型进行了比较。模型得到的解析共振频率为19378Hz，与数值模型的19270Hz和实验测试的19457Hz相比，误差分别为0.4%和0.5%。数学模型得到的换能器尖端在谐振频率下的振幅为21.2微米，比实验测试得到的22微米低3.6%。数学模型计算出的反谐振频率为21748Hz，其值为0。