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研究生:張文曲
研究生(外文):Wen-Chey Chang
論文名稱:使用波傳方法之多跨距樑振動研究
論文名稱(外文):VIBRATION OF MULTIPLE-SPAN BEAMS USING WAVE PROPAGATION METHOD APPROACH
指導教授:王輝清
指導教授(外文):Hui-Ching Wang
學位類別:博士
校院名稱:國立中興大學
系所名稱:應用數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:中文
論文頁數:174
中文關鍵詞:振動波傳轉換矩陣穿透波反射波散射矩陣奇異值分解
外文關鍵詞:VibrationWave PropagationTransformation MatrixTransmission WaveReflection WaveScattering MatrixSingular Value Decomposition
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本文主要以兩個不同的波傳方法研究多跨距樑的動態響應。首先,利用在支撐點的位移、旋轉連續與彎矩和剪力的平衡條件,輕易使用狀態向量轉換的方法得到多跨距樑的動態響應方程式,再代入邊界條件後可獲得特定跨距的動態響應,此特定跨距將成為整體樑結構動態響應之縮影。其次,再以波傳的散射觀念,並利用波的穿透與反射現象,獲得在作用力處包含支撐點與邊界之散射矩陣方程式,將可提供研究多跨距樑的動態特性以及模態分析。兩種方法均運用奇異值分解的方法進行奇異值分解,並以數個計算例子證明奇異值與多跨距樑的動態特性之關連性,藉以提供多跨距樑隔振分析。

The research is focused on dynamic response for a finite multi-span continuous beam. Two different formulations are presented. First, a formulation is using transformation of state vectors, which is defined as displacement, rotation, moment, and shear force at supports. The state vector variation across a support is represented by transformation matrix. This method is easy to relate the state vector of every span by using transformations. There are four unknown variables in the governing equation, which are the components of propagating and non-propagating waves traveling to the left and to the right directions. Imposing the boundary conditions yields the equation for the multi-span beam of which the entire beam’s dynamics is condensed in one selected span. The present method is employed to analyze vibration isolation of a finite multi-span beam. Secondly, a scattering matrix is introduced to describe the transmission and reflection phenomena of which two incident waves from two different directions impinge upon a support or multiple supports. Reflection matrices specify the boundary conditions of the beam. The waves generated at the position of applied force are two outgoing waves, which in turn generate two reflected waves caused by the boundaries. These two reflected waves are derived based on the scatter matrices of the supports and the reflection matrices of the boundaries. The proposed formulation is used to study the dynamic characteristics of a multi-span beam and to develop the associated modal analysis. In both formulations a Singular Value Decomposition (SVD) analysis is applied to the equation to study the dynamics of the beam in terms of the eigenvectors associated with the singular values. Numerical examples are provided to demonstrate the formulation and associated dynamic characteristics as represented by the SVD analysis.

誌 謝 ii
ABSTRACT iii
摘 要 v
符號說明 xiv
第1章 緒論 1
1.1 前言 1
1.2 文獻回顧 2
1.3 研究方法 11
1.4 本文架構 13
第2章 狀態向量轉換之多跨距樑振動分析 15
2.1 單跨距Euler-Bernoulli樑之統御方程式 15
2.2 單跨距與支撐點之狀態向量轉換 16
2.2.1 單跨距狀態變數之轉換矩陣 17
2.2.2 支撐點狀態變數之轉換矩陣 19
2.3 多跨距樑之狀態向量轉換 21
2.4 振幅係數與狀態向量之求解 24
2.5 奇異值分解法之應用 28
2.6 半無窮樑之理論推導 30
2.6.1 右端半無限樑 30
2.6.2 左端半無限樑 34
第3章 行波在多跨距樑之反射與傳遞分析 37
3.1 單一支撐點的散射波 37
3.2 多點支撐之多重散射 42
3.3 邊界條件之反射矩陣 45
3.4 多跨距樑的響應理論 47
3.5 行波在角度樑之反射與傳遞 53
3.5.1 縱向應力波的理論基礎 53
3.5.2 多重折點之多重散射 54
3.5.3 方形框架結構的響應理論 56
第4章 數值計算例子與討論 61
4.1 狀態向量轉換之三跨距樑 61
4.2 狀態向量轉換之半無窮邊界條件 68
4.3 狀態向量轉換之三跨距樑的支撐阻尼效應 69
4.4 波的傳遞與反射之振動分析 71
4.5 多跨距樑有限元素法之模態響應比較 73
第5章 結論和未來研究方向 75
5.1 結論 75
5.2 未來研究方向 78
附錄 79
參考文獻 82
自述 159

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