應用波傳元素分析圓管結構的振動

Abstract

本文以二維波傳元素推導圓管結構的頻率方程式。軸對稱圓管結構上的駐波包括扭轉模式、縱向模式及撓曲模式，前兩者具有軸對稱位移，以軸向正傳、逆傳之指數波傳因子作為元素的內差函數。撓曲振動具有較複雜的的位移，需藉助周向匝數n及軸向模態數m予以區分，除了指數波傳因子外，還需要以周向位置變數之周向匝數倍角三角函數作為內差函數。 本文另以套裝有限元素軟體ANSYS進行共振模態分析，採用實體元素solid 45及薄殼元素shell 63，探討不同直徑、軸向模態數之圓管共振頻率數值解的收斂性。在低周向匝數時，兩種元素在各軸向模態數下，均可以達到收斂。隨著n、m的增加，高階模態對於元素數目逐較敏感。薄殼元素不需進行厚度分割，在薄壁圓管分析上有較佳的收斂性。 圓管結構撓曲振動的共振頻率不依照周向匝數或軸向模態數之大小，依序高低排列。本文採用兩個無因次因子：頻率因子□、軸向波長因子□，探討圓管結構共振頻率的變化趨勢。周向匝數n □ 2時，頻率因子隨著軸向波長因子的增大而增加，兩者的比例關係隨著n值的持續增加而趨於下降。當軸向波長因子小於1.5時，頻率因子隨著n值的增加而單調遞增。超過此範圍，頻率因子隨著n值的增加，呈現先減少再增加的變化趨勢。

A formulation of two-dimensional wave elements for frequency equations of circular hollow structures is developed in this thesis. The standing waves appearing in a cylindrical tube comprise torsional modes, longitudinal modes, and flexural modes. The former two kinds of modes have axially symmetric motions. The exponential propagators in forward and backward axial directions are used as high order interpolation functions in each element. Flexural modes have more complicate motions and could be categorized by circumferential number n and axial mode number m. Besides the exponential propagators, flexural motions also depend on the circumferential angle through the trigonometric functions defined by n. Further calculations for resonant frequencies and modes of vibration were carried out by a commercial finite element code ANSYS with Solid 45 and Shell 63 elements. The convergence tests for resonant frequencies with respect to a variety of diameters of cylinder and axial modes were performed. In lower circumferential numbers, both elements can reach convergence in a wide range of axial mode numbers. But the converging results for higher order modes are more sensitive to the number of elements. Better convergence could be achieved by shell elements because of no meshing in thickness. The resonant frequencies of flexural vibration do not appear in sequence of the circumferential number or the axial mode number. Two non-dimensional parameters, frequency factor □ and axial wavelength factor □, are adopted to explore the variation of resonant frequencies for flexural vibration with various numbers of n and m. Numerical results indicate that the frequency factor increases with axial wavelength factor for the circumferential number less than 2. But the proportionality might become less significant with the increase of n. The frequency factor increase monotonically with the axial mode number in the range less than 1.5. Beyond this range, it decreases first and is followed by increasing with the axial mode number.