線型與圓柱型聲波導的雙維有限元素分析

Abstract

本文應用漢彌頓原理與有限元素法，以雙維有限元素法分析線型及圓柱型聲波導的頻散特性及共振模態。另以ANSYS三維有限元素法分析圓柱楔形體，計算各模態的共振頻及對應之導波相速度頻散曲線，並探討元素的收斂性。以有限元素法三維分析圓柱型聲波導，受限於元素分割數目，只能分析較低次的共振模態。雙維有限元素法的聲波導分析基礎於分離變數法，將導波的時諧波傳因子與截面振動分離成雙維有限元素，可以將分析範圍擴及高頻及高次共振模態，並減少分割元素的數目，計算精度也大幅提昇。雙維有限元素分析結果顯示，線型楔型體聲波導反對稱波的模態會隨著頂角角度增加而減少，且波速小於芮利波波速，與Lagasse經驗公式的結果一致。線型與圓柱楔形體聲波導波頻散曲線，受底部固定之邊界條件影響，在低頻範圍內相速度會明顯拉高。邊界條件對於楔型體聲波導的影響只發生在低頻的波段，結構曲率對楔型聲波導的影響，則隨著模態愈高次而愈明顯。線型及圓柱楔形體聲波導的導波能量集中在頂角，且為反對稱撓性波的型態。而線型長方體及圓柱長方形，由於斷面型態無尖角，波的能量分佈於整個斷面，除了反對稱的撓性波外，另外還會有延性波等不同型式的振動模態出現。

A bi-dimensional finite element model based on Hamilton’s principle and finite element method is developed in this thesis to analyze the dispersive characteristics and mode shapes of normal modes for linear and circular cylindrical acoustic waveguides. The dispersion curves of phase velocities for guided waves and their corresponding resonant frequencies for a circular-wedge waveguide were also evaluated by 3D finite element analysis (FEA) using the commercial code, ANSYS ver.5.7. The convergence was simultaneously discussed. The 3D FEA has limitation in calculation for higher normal modes due to constraint in the available number of elements. The bi-dimensional finite element method is based on separation of variables, in which the wave propagation factor is separated from cross-sectional vibrations of the acoustic waveguides. The present method has advantages in determination of phase velocities and mode shapes up to higher normal modes and in a wide range of frequencies without loss of accuracy. Phase velocities of the antisymmetric flexural (ASF) guided waves in linear-wedge waveguides are found to be slower than the Rayleigh wave speed. The calculated results in the range of higher wave numbers are in a good agreement with the empirical formula provided by Lagasse. The ASF waves in either linear or circular cylindrical wedge-typed waveguides have faster and frequency-dependent phase velocities in the range of lower wave numbers. It results from the boundary conditions on the bottom of waveguides, which are different from the ideal wedge problem considered in Lagasse’s work. In addition, curvatures of the acoustic waveguides increase the phase velocities of higher normal modes only. Contrary to the wedge-typed waveguides, the guided wave propagation in both linear and circular rectangular waveguides is dispersive. Most energy carried by the ASF waves in the wedge-typed waveguides is confined at the tip of wedge and is observed in the corresponding mode shapes. However, wave motion for the rectangular waveguides spreads over whole cross section. The evidence indicates that other kinds of normal modes such as extensional waves appear in the rectangular waveguides.