一种弯张换能器及其共振频率计算方法

A flextensional transducer and its algorithmic method of resonant frequency

提出一种弯张换能器即欧米伽换能器,推导出其共振频率和位移振形函数。把欧米伽换能器分成四个构成部分,利用旋转薄壳理论和压电理论分别求出各部分的能量并进行相加,得到整个欧米伽换能器能量的泛函表达式;把几何边界连续条件和包含待定系数的位移振形函数代入到欧米伽换能器能量泛函中,运用Rayleigh-Ritz法求出欧米伽换能器的共振频率,再把共振频率代入Rayleigh-Ritz偏微分方程和边界方程中求出位移振形函数的待定系数以获得确定的位移振形函数。该方法也被推广到对钹式换能器共振频率和位移振形函数的求解上。上述求解结果与实验结果和数值软件相结合论证了该方法的有效性。可获得以下结论:(1)欧米伽换能器陶瓷片的径向振动与金属端帽顶部的纵向振动同相,减少了声场中的反相分量,易作为低频换能器使用;(2)为解决欧米伽换能器和钹式换能器的优化设计提供了理论支持;(3)文中求解共振频率和位移振形函数的方法,即可以应用在由旋转薄壳组成的弯张换能器上也可以应用在由旋转薄壳组成的其它结构上,具有普遍性。

A flextensional transducer with an Omega shape and its algorithmic method of the resonant frequency and the shape functions are suggested. The Omega transducer is separated into four parts, each part is treated as a thin shell of revolution, and the theories of thin shells of revolution and piezoelectricity are used to obtain the energy functional of each part. The sum of the energy functionals of the four parts is the energy functional of the whole Omega transducer. By substituting the displacement shape functions with undetermined coefficients and the geometrical boundary conditions into the energy functional of the Omega transducer, the resonant frequency of the Omega transducer is firstly determined with the Rayleigh-Ritz method. With the gotten resonant frequency, the constant coefficients of the displacement shape functions are following solved through the Rayleigh-Ritz partial differential equations and the geometrical boundary conditions equations. The solving method of the resonant frequency and displacement shape functions is also extended to the Cymbal transducer. Such an analytical method is verified to be feasible by the results of the Ansys numerical calculation and experiments. The reasearch indicated： （1） the radial vibration of the piezoelectric ceramics is in phase with the longitudinal vibration of the top of metal cap. The Omega transducer can be a low frequency transducer. （2） The determination method of the resonant frequency and the displacement shape functions give a solution in the optimum designs of the Omega transducer and the Cymbal transducer. （3） The determination method of the resonant frequency and displacement shape functions can also be used in other flextensional transducers or other structures which are composed of thin shells of revolution.