纵向换能器的频率方程

Frequency equations for a longitudinal transducer

目前应用的纵向换能器频率方程是通过1个未知节点分成2个部分分别描述的,该方程必须借助数学迭代方法计算整体换能器的谐振频率。针对其应用的不便性和对不同截面形状的适应性问题,本文基于一维振动理论和能量法,给出纵向换能器的简化机电等效模型,考虑部件截面参数,通过动能和位能等效关系得出等效元件参数表达式,根据谐振条件经过严格数学推导,给出关于整体换能器频率方程的参数化数学表达式。可应用于部件不等截面以及多种截面形状尺寸的纵向换能器谐振频率和振动节点计算,具有更广泛的适用性,计算结果与有限元方法符合很好。

Currently used frequency equations for a longitudinal transducer are described as two segments from an unknown node,which must employ the mathematical iterative method to calculate the resonant frequency of the entire transducer.Therefore,frequency equations are inconvenient and have adaptability problems on transducers with different cross-sectional shapes.In this study,we obtained the simplified electromechanical equivalent model on the basis of the one-dimensional vibration theory and energy method and deduced the parametric expression of equivalent elements through the equivalent relationship between kinetic and potential energies,considering the parameters of cross sections.Finally,on the basis of the mechanical resonance condition,we derived the parameterized mathematical expression of frequency equations for the entire transducer,which is precisely deduced through the mathematical method.This parameterized mathematical expression can be widely used to calculate the resonant frequency and vibrational node of longitudinal transducers with unequal cross-sectional parts or different cross-sectional shapes and sizes.The calculated results are consistent with those of the finite element method.