臺灣博碩士論文加值系統

本文目的在利用數值組合法(numerical assembly method，簡寫為 NAM)來求解一均勻或不均勻樑攜帶一組或多組集中元素(各組包含一具有偏心距 及轉動慣量 之集結質量 、一線性彈簧 與一螺旋彈簧 )承受軸向力 作用時的自然頻率與振態。首先，吾人將一不均勻Euler-Bernoulli樑分割為數根段樑(beam segments)，將每相鄰的兩根段樑以一節點連接之，並將上述各種集中元素附著於各個節點上。然後推導一典型段樑(typical beam segment)承受軸向力作用時的位移函數。接著，根據樑上每個節點處的位移與斜率之相容方程式、力與彎矩之平衡方程式、以及與整根樑兩端點的邊界有關之邊界條件方程式，吾人可得一組聯立方程式。將上述方程式寫成矩陣的形式，則得一特徵方程式(characteristic equation)，令其係數行列式等於零，可得一頻率方程式，解之，則得整個振動系統的自然頻率；將各個無因次化自然頻率常數，代入上述的特徵方程式，即得各個相關段樑的積分常數，利用這些積分常數與各段樑的位移函數，吾人即可獲得對應的振態。為驗証本文理論與電腦程式之可靠性，吾人將本文結果與現有文獻所得的結果比較，因所有的相關數據皆非常接近，故本文理論與電腦程式的可靠性應可被接受。

This thesis determines the natural frequencies and the corresponding mode shapes of an axial-loaded uniform or non-uniform beam carrying any sets of concentrated elements by using the numerical assembly method (NAM) with each set of concentrated element consisting of a lumped mass with eccentricity and rotary inertia, a translational spring and a rotational spring. First of all, a uniform or non-uniform beam is subdivided into many beam segments with any two adjacent beam segments joined at a node and each node is attached by one set of aforesaid concentrated element. Next, the displacement function of a typical axial-loaded beam segment is derived. By using this displacement function and incorporating with the compatible equations of displacements and slopes and the equilibrium equations of forces and moments at each intermediate node, along with the equations concerning the boundary conditions of the entire beam, one may obtain a set of simultaneous equations. Writing the last equations in matrix form, one obtains the characteristic equation of the vibrating system and setting its coefficient determinant to be zero, one obtains the frequency equation. Finally, one may determine the natural frequencies of the title problem from the frequency equation and the associated integration constants of all beam segments by substituting each of the natural frequencies into the last characteristic equation. Based on the last integration constants and the displacement functions for all beam segments, one may obtain the mode shape corresponding to each natural frequency. Based on the good agreement between the results of this thesis and those of the existing literature, it is believed that the reliability of the theory presented and the computer program developed for this thesis should be acceptable.

摘要............................................I
Abstract........................................II
誌謝............................................IV
目錄............................................V
表目錄..........................................VII
圖目錄..........................................IX
符號說明........................................XII

第一章 緒論.....................................1
1-1 研究動機....................................1
1-2 文獻回顧....................................2
1-3 研究方法....................................5

第二章 理論分析.................................6
2-1 段樑的運動方程式與位移函數..................7
2-2 整個振動系統的邊界條件......................10
2-3 整個振動系統的頻率方程式....................11
2-3-1 整根樑只被分割為兩根段樑時的頻率方程式..14
2-3-2 整根樑被分割為三根段樑時的頻率方程式....18

第三章 數值分析結果與討論.......................24
3-1 均勻Euler樑的自然頻率與振態.................24
3-2 均勻Euler樑承受軸向壓力時的自由振動頻率.....30
3-3 兩端自由的均勻樑承受軸向力時的自由振動分析..35
3-4 不均勻Euler樑承受軸向力時的自由振動分析.....42
3-5 軸向壓力對傾斜樑第一自然頻率的影響..........51
3-5-1 固定厚度的傾斜樑........................51
3-5-2 圓錐型傾斜樑............................58
3-6 附帶多種集中元素之均勻與不均勻樑承受軸向壓力
作用時的自由振動析..........................65

第四章 結論.....................................81

參考文獻........................................83

附錄A 電腦程式使用說明..........................86

自述............................................96

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