The Langevin transducer is a widely used power ultrasonic device across both medical and industrial applications, from orthopaedic surgery to drilling and welding. It is a sandwich-type device which typically consists of piezoelectric ceramic rings between two metallic end-masses. The transducer is commonly operated at a particular resonant mode to deliver ultrasonic vibrations to a target structure of interest, and is generally modelled using approaches including the transfer matrix method and electromechanical equivalent circuit methods. Here, we propose a variational framework to study the dynamics of the Langevin transducer based on the Timoshenko-Ehrenfest beam theory and Hamilton's principle. The variational equation derived from this model is then discretized by the standard finite element method with spectral elements. To verify the proposed one-dimensional electromechanical model, the computed resonance frequencies, or natural frequencies, of the one-dimensional model are compared to those of a three-dimensional finite element model with respect to varying geometry parameters characterizing the transducer. The results of the reduced one-dimensional and full three-dimensional models are then compared to those measured through an experimental investigation consisting of laser Doppler vibrometry. This is undertaken for the first ten eigenfrequencies, including longitudinal and bending modes, where normalized amplitudes and vibration mode shapes are reported. The close correlation between modelling and experiment demonstrates that the proposed one-dimensional electromechanical model can deliver results consistent with the full three-dimensional model and from experiment, thus verifying a rapid and reliable method for studying the free vibrations and dynamics of the Langevin transducer which accounts for the axial vibrations of the piezoelectric ceramic stack in all cases.

中文翻译：

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用于模拟 Langevin 换能器动力学的 Timoshenko-Ehrenfest 梁模型

Langevin 换能器是一种广泛应用于医疗和工业应用（从骨科手术到钻孔和焊接）的功率超声波设备。它是一种三明治型装置，通常由两个金属端块之间的压电陶瓷环组成。换能器通常在特定谐振模式下操作，以将超声波振动传递到感兴趣的目标结构，并且通常使用包括传递矩阵方法和机电等效电路方法在内的方法进行建模。在这里，我们提出了一个基于 Timoshenko-Ehrenfest 梁理论和哈密顿原理来研究 Langevin 换能器动力学的变分框架。然后使用谱元素通过标准有限元方法对从此模型导出的变分方程进行离散化。为了验证所提出的一维机电模型，将一维模型的计算谐振频率或固有频率与三维有限元模型的谐振频率或固有频率相对于表征换能器的变化几何参数进行比较。然后将简化的一维和全三维模型的结果与通过激光多普勒测振法​​组成的实验研究测量的结果进行比较。这是针对前十个特征频率进行的，包括纵向和弯曲模式，其中报告了归一化振幅和振动模式形状。建模与实验之间的密切相关性表明，所提出的一维机电模型可以提供与全三维模型和实验一致的结果，从而验证了一种快速可靠的研究 Langevin 换能器自由振动和动力学的方法。在所有情况下都考虑了压电陶瓷堆栈的轴向振动。