壓電圓管脊緣撓性波的周向波傳

Abstract

圓管脊緣撓性波的波傳問題迄今尚無確切解，通常仰賴數值解析或經驗公式。本文探討壓電圓管脊緣撓性波的頻散特性與共振模態，以分離變數法將圓柱體截面的位移表示成截面座標的函數與周向波傳因子的乘積，以雙維有限元素法與漢彌頓原理推導脊緣撓性波的頻散方程式，數值解析行進波的頻散曲線及駐波的共振頻率，應用於超音波馬達的最佳化設計及相鄰共振頻率之模態隔離。 實驗方面以網路分析儀量測兩端自由之壓電圓管的阻抗曲線，將共振頻率量測值與脊緣撓性波的頻散曲線比較，驗證數值解析的正確性。本文以簡單體法反算壓電圓管的幾何參數與彈性係數，策略性的先反算前者，在據以反算後者，有效地提高彈性係數反算的準確性。採用高階軸向模態的共振頻率量測值，可以減少目標函數局部極小值的發生。

Flexural ridge wave propagation around a circular cylindrical tube has no exact solution up to the present. It is used to be solved numerically or using an empirical formula. This thesis investigates the dispersive properties of ridge waves traveling circumferentially around the piezoelectric tubes and their resonant modes. Based on separation of variables, the displacements of ridge wave are represented as the product of a cross-sectional coordinate depending function and the propagator along the circumference of tube. The dispersion equation of ridge waves is formulated by using Hamilton’s principle and so-called bi-dimensional finite element method. Dispersion curves of traveling waves and resonant frequencies corresponding to standing waves are solved numerically. Several applications are illustrated such as optimal design for ultrasonic motors and modal separation of structure among adjacent resonant frequencies. The impedance curves of a free-free piezoelectric tube are measured by a network analyzer. The measured resonant frequencies are compared with the predicted dispersion curves of ridge waves. Validity of the present numerical approach has been verified. Geometric parameters and elastic constants of the piezoelectric tube are determined through an inverse scheme based on the measured resonant frequencies by using simplex method. A good accuracy in inversion for elastic constants can be strategically achieved by determining the geometric parameters first. Then it is followed by seeking the best likelihood of elastic constants. The number of local minima during inversion can be reduced if more measured resonant frequencies of higher order axial modes are included in the objective function.