Vibration analysis of a cracked simply supported beam with a moving load of constant speed 顏肇賢、林海平
E-mail: [email protected]
ABSTRACT
In this study, a hybrid numerical/analytical method that permit the efficient calculation of dynamic characteristics of a cracked simply supported beam with a moving load of constant speed is presented. First, assuming the beam obeying the Euler-Bernoulli beam theory, the equation of motion of the system is derived. By using the transfer matrix method, eigensolutions (natural frequencies and mode shapes) of the cracked system can be determined. Forced response of the moving load can be obtained by model expansion theory. Some numerical results are calculated and compared with the previous researches.
Keywords : Euler-Bernoulli, transfer matrix, eigensolutions, model expansion theory Table of Contents
目錄 封面內頁 簽名頁 授權書 iii 中文摘要 v 英文摘要 vi 誌謝 vii 目錄 viii 圖目錄 x 表目錄 xii 符號說明 xiii 第一章 緒論 1 1.1 前言 1 1.2 文獻回顧 2 1.2.1 有關含破壞點之Euler-Bernoulli 樑理論的文獻 3 1.2.2 有關含破壞點與移動負荷之Euler 樑理論的 文獻 5 1.3 研究目標 7 1.4 本文架構 9 第二章 研究方法 10 2.1 分析研究步驟 11 2.2 Euler-Bernoulli 樑之運動方程式[1][2] 12 2.3 含有破壞點之Euler-Bernoulli 樑[11-13] 16 2.4 樑之各種邊界條件之介紹[1][2][3][17] 18 2.5 破裂處情形之介紹[7] 20 2.6 計算單破壞點之簡支樑系統之特徵值[11][12] 22 2.6.1 變數變換處理 22 2.6.2 變數分離處理 23 2.6.3 計算特徵值 25 2.7 其他 不同形式之邊界矩陣與特徵值[17] 29 2.8 樑側向振動之強制振動分析[1][2] 35 第三章 移動負荷分析 38 3.1 分析具單一等速 移動負荷之損傷樑 38 3.2 分析具兩個等速移動負荷之破壞樑 42 3.3 側向位移響應的收斂性情形與r 值的關係 47 第四章 數 值分析結果與討論 49 4.1 理論數值分析與文獻互相驗證 49 4.2 單一移動負荷與兩個移動負荷之理論數值分析 52 4.2.1 單一 移動負荷之理論數值分析 53 4.2.2 兩個移動負荷之理論數值分析 65 第五章 結論與建議 69 5.1 結論 69 5.2 建議 71 參考文獻 72 圖目錄 圖2.1 研究進行步驟流程圖 11 圖2.2 Euler-Bernoulli 樑上截面積變化之示意圖[1] 12 圖2.3 樑元素之自由體圖平衡 狀態[1] 13 圖2.4 具破壞點之Euler-Bernoulli 樑之示意圖[13] 16 圖2.5 一般常見之邊界情形[17] 18 圖2.6 各種邊界情形所考量 的因素[17] 19 圖2.7 破裂點在樑之單側[7] 21 圖2.8 破裂點在樑之兩側[7] 21 圖2.9 具固定負荷與破壞點之Euler-Bernoulli 樑 之示意圖[1] 35 圖3.1 具單一移動負荷與破壞點之Euler 樑之示意圖 38 圖3.2 兩軸車輛橫過單一破壞點之Euler 樑之示意圖 43 圖3.3 具兩個移動負荷與破壞點之Euler 樑之示意圖 43 圖3.4 系統側向位移響應的收斂性情形 48 圖4.1 速度影響之下均勻 樑中間點的時間位移響應；moving load ( γ=0)， moving load ( γ=0.5)[22] 50 圖4.2 不同速度之下，均勻樑中間點的時間位 移響應，實線代表/ 0 a H = ，虛線代表/ 0.5 a H = 52 圖4.3 不同速度下，破裂深度改變，均勻樑中間點時間位移響應，破 裂位置位1 0.3 L = m 57 圖4.4 不同速度下，破裂深度改變對，均勻樑中間點的時間位 移響應，破裂位置位於1 0.5 L = m 58 圖4.5 不同破裂深度下，改變移動負荷的速度，均勻樑中間點的時間位移響應，破裂位置位於1 0.3 L = m 62 圖4.6 不同破 裂深度下，改變移動負荷的速度，均勻樑中間點的時間位移響應，破裂位置位於1 0.5 L = m 63 圖4.7 不同的破裂深度和移 動負荷速度之下，改變其破裂點的位置，其均勻樑中間點的時間位移響應，實線代表1 0.3 L = m，虛線代表1 0.5 L = m 64 圖4.8 破裂點的位置位於1 0.5 L = 公尺，破裂深度比值是/ 0.5 a H = 時，改變負荷間的距離，其均勻樑中間點的時間位移 響應，實線代表 /8 D L = ，虛線代表/ 4 D L = 66 圖4.9 破裂點的位置位於1 0.5 L = 公尺，破裂深度比值是/ 0.5 a H = 時
，改變負荷間的距離，其均勻樑中間點的時間位移響應，實線代表 /16 D L = ，虛線代表/8 D L = 66 圖4.10 破裂點的位置 位於1 0.5 L = 公尺，破裂深度比值是/ 0.5 a H = 和移動負荷速度分別在0.2、0.4 的臨界速度時，兩個移動負荷( /8 D L = ) 與單一移動負荷的側向位移響應，實線代表單一移動負荷，虛線代表兩個移動負荷 67 圖4.11 破裂點的位置位於1 0.5 L = 公尺，破裂深度比值是/ 0.5 a H = 和移動負荷速度分別在0.6、0.8 的臨界速度時，兩個移動負荷( /8 D L = )與單一移動負 荷的側向位移響應，實線代表單一移動負荷，虛線代表兩個移動負荷 68 表目錄 表2.1 利用Euler 樑理論時，不同邊界情形 所考量的因素[1][2] 20 表4.1 數值分析之材料幾何特性與材料特性(一)[22] 50 表4.2 數值分析之材料幾何特性與材料特性(
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