臺灣博碩士論文加值系統

一般而言，對於同樣的問題，最好有許多種不同的解法，唯有如此，面對各種不同的情況，吾人才有較多的選擇，因此，本文之主旨在於提出三種方法，來求解頂端攜帶一偏心且具有轉動慣量之集結質量的(部份或全部)浸水均勻樑的自然頻率及振態。其中，方法一是傳統的解析法，本法直接從整個振動系統的運動方程式與相關的邊界條件，在無額外假設的情況下，利用純粹的數學運算，而求得整個振動系統的自然頻率及振態，由於在求解過程中，吾人並未將振動系統離散化，故所得結果屬於正解(exact solutions)，可作為其他近似法準確度的評估基準(benchmark)。方法二是解析與數值混合法(ANCM)，本法之要點有三，首先是利用解析法求得(無浸水)乾樑的自然頻率及對應振態之解析解(analytical solution)；其次是利用上述乾樑的自然頻率及振態，以及振態重疊法的理論，將(浸水)濕樑的偏微分方程式，轉換成矩陣方程式；最後是利用各種數值法，來求解(浸水)濕樑的自然頻率及振態。依不同的裸樑(bare beam)與負荷樑(loaded beam)模式，本法又有兩種解法，本文稱之為方法二(A)及方法二(B)。由於解析與數值混合法(ANCM)只有在裸樑(bare beam)的振態具有正交性時才適用，故不同模式的裸樑之振態正交性條件(orthogonality conditions)，是本方法二的研究重點。方法三是傳統的有限元素法(FEM)，本法之基本概念是將樑浸水部份的附加質量(added mass)，用許多集中質量(concentrated masses)取代之，然後將一(浸水)濕樑視為一攜帶許多集中質量的(無浸水)乾樑，接著，利用限元素法的組合技巧，則可獲得整個振動系統的總質量矩陣及總勁度矩陣，最後，本文是以Jacobi法來求解特徵值方程式，而得到(浸水)濕樑的自然頻率及振態。比較數值分析結果，吾人發現上述三種方法所得的數據皆非常接近，這表示本文所提理論與所發展的電算程式之可靠性，應可被接受。此外，吾人亦曾進行一簡單的模型試驗，結果顯示，實驗量測數據與理論計算數據亦非常接近。

In general, only if there exist many techniques for solving the same problem, then one has the best choice to incorporate with various situations. For the last point of view, this thesis aims at presenting three methods for the free vibration analysis of an immersed uniform beam carrying an eccentric tip mass with rotary inertia. Method 1 is a classical analytical method. It directly determines the natural frequencies and mode shapes of the vibrating system from the equation of motion and the associated boundary conditions. Because the continuous model instead of the discrete one is solved, the solutions obtained from Method 1 are the exact ones and may the benchmark for evaluating the accuracy of the other approximate methods. Method 2 is an analytical-and-numerical-combined method (ANCM). The key points of this method are: (i) Determine the analytical (exact) solutions for the natural frequencies and mode shapes of the “dry” beam (without contacting with water); (ii) Transform the partial differential equation of the “wet” beam (contacting with water) into a matrix equation by using the last natural frequencies and mode shapes of the “dry” beam and the mode superposition theory; (iii) Solve the last matrix equation to give the natural frequencies and mode shapes of the “wet” beam. According to different definitions for the “bare” beam and “loaded” beam, the current Method 2 may be subdivided into two approaches, they are called Method 2(A) and Method 2(B) respectively, in this thesis. Because ANCM is available only if the mode shapes of the bare beam are orthogonal to each other, the derivation of orthogonality conditions for the “dry” beam is the main task of Method 2. Method 3 is the conventional finite element method (FEM). The basic concept of this method is to replace the added mass of the immersed part of the beam by many point masses, and then a “wet” beam is equivalent to a “dry” beam carrying a lot of point masses. Next, the assembly technique of FEM is used to establish the overall mass and stiffness matrices of the entire vibrating system. Finally, the Jacobi method is used to solve the characteristic equation to yield the natural frequencies and mode shapes of the “wet” beam. Numerical results reveal that the natural frequencies and mode shapes of the “wet” beam obtained from the above-mentioned methods are very close to each other. This confirms the reliability of the theories presented and the computer programs developed for this thesis. Furthermore, a simple model test is also conducted and good agreement between the results of experiments and those of theoretical calculations is achieved.

摘 要 I
Abstract III
目 錄 VI
表 目 錄 VIII
圖 目 錄 X
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 8

第二章 以解析法求浸水樑之自然頻率及振態(方法一) 10
2-1基本假設 10
2-2浸水樑的自然頻率及振動模態 12

第三章 解析與數值混合法(ANCM)(方法二) 18
3-1 “乾樑”之自然頻率及振態--方法二(A) 19
3-2 “乾樑”之自然頻率及振態--方法二(B) 25
3-3振態的正交條件(orthogonal conditions)方程式 28
3-4濕樑之運動方程式及其解--方法二(A) 32
3-5濕樑之運動方程式及其解--方法二(B) 37

第四章 有限元素法(FEM)(方法三) 41
4-1有限元素法(FEM)之數學模型(Mathematical model) 41
4-2 樑元素之質量矩陣與勁度矩陣 43

第五章 數值分析結果與討論 45
5-1 理論與電腦程式之可靠性 47
5-2 方法二(A)與方法二(B)之準確性 49
5-3 底端彈性支撐浸水樑彈簧勁度(kT=kR)之影響 55
5-4 頂端質量偏心距(eccentricity)之影響 59
5-5 頂端質量比及轉動慣量(rotary-inertia)比之影響 61
5-6 具跨距堆積質量(in-span lumped mass)之影響 65

第六章 實驗與結果 69

第七章 結論 74

參考文獻 76
附錄A: 方法二(A)與方法二(B)之正規化因子(normalization factor) 81
附錄B: 樑元素之質量矩陣與勁度矩陣 83
自述 86

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