一根楔形樑在空氣中(本文稱為乾樑)之自然頻率與其浸在水中(本文稱為濕樑)之自然頻率不一樣，係由於水的影響。因此，假如吾人在計算乾樑的自然頻率及相對應的振態時，有考慮到濕樑振動時所牽涉到的附加質量(added mass)效應，則所得之自然頻率及振態應該會與濕樑相等。根據上述的觀念，本文使用兩種方法來求楔形樑在水中振動的自然頻率及相對應之振態：有限元素法(finite element method；FEM)及點附加質量法(point added mass method ;PAM)。在使用有限元素法解題時，吾人將樑浸水部份的長度等分成 個樑元素，然後使用附加質量係數 來估算每一樑元素的附加質量 ，再將 平均分配到各相關樑元素之兩端節點(nodes)上，而以集中質量之方式附加到乾樑元素之相關節點上，以求得乾樑在水中振動時的自然頻率及相對應之振態。在使用點附加質量法解題時，吾人先使用Bessel函數來求楔形樑之頂端未附帶集中質量時在空氣中振動之自然頻率及正規化振態。接著，吾人將浸水部份的各個附加集中質量與樑頂端所附帶之實際集中質量所產生的慣性力(inertial forces)施加到上述的乾樑上，並利用擴展理論，來推導相關的特徵值方程式。後者之特徵值與特徵向量即為頂端附帶一集中質量之乾樑在水中振動之自然頻率及振態。

It is well-known that, due to effect of the surrounding water, the natural frequencies of a beam in air (or dry beam) are different from those of the same beam immersed in water (or wet beam). However, if the natural frequencies and the associated mode shapes of a dry beam are calculated by taking account of the “added mass” for the immersed beam, then the last natural frequencies and mode shaps will be equal to the corresponding ones of the wet beam. Based on the last concept, the closed form solutions for natural frequencies and the associated mode shapes of the dry beam were determined first, then the partial differential equation of motion for the wet beam was transformed into a matrix equation by using the expansion theorem and the foregoing closed form solutions of free vibration responses for the dry beam. Solving the last matrix equation will give the required natural frequencies and the associated mode shapes of the wet beam. The formulation of this paper is available for the fully or partially immersed double tapered beams with either circular, square or rectangular cross-sections. The taper ratio for width and that for depth may be equal or unequal. The numerical results of this paper were compared with the existing results or the finite-element-method results and good agreement was achieved.

Keywords: Dry beam; Wet beam; Added mass; Natural frequencies; Mode shapes; Circular or rectangular cross-sections

摘要…………………………………………………………I
誌謝………………………………………………………II
目錄………………………………………………………III
表目錄………………………………………………………V
圖目錄……………………………………………………VI
符號說明…………………………………………………VII
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 有限元素分析…………………………………… 6
2-1 基本假設………………………………………………6
2-2 樑元素之質量矩陣與勁度矩陣………………………7
2-3 有限元素模型…………………………………………9
2-4 乾樑頂端附帶一集中質量之元素性質矩陣……… 10
2-5 濕樑頂端附帶一集中質量之元素性質矩陣……… 11
第三章 點附加質量法……………………………………13
3-1乾樑的運動方程式及其自然頻率與振態……………13
3-2 頂端附帶一集中質量之濕樑的自然頻率與振態… 18
第四章 數值分析結果與討論……………………………23
4-1 理論與電腦程式的可靠性………………………… 23
4-2 浸水的效應………………………………………… 27
4-3 點附加質量法(PAM)的適用性………………………30
第五章 結論………………………………………………33
參考文獻………………………………………………… 34
附錄A：乾樑振態之閉式解析解…………………………39
附錄B：Bessel函數的相關公式 …………………………42
附錄C：樑元素之質量矩陣與勁度矩陣…………………44
自述……………………………………………………… 47

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