臺灣博碩士論文加值系統

由陳長鈕教授所研究開發出來的數值積分表示微分元素法為一種分析連體力學問題的數值方法。數值積分表示微分元素法是將欲分析的結構物分割成有限個元素，在利用數值積分表示微分的技巧，對定義各個元素內的統御微分或偏微分方程式，兩個相鄰的元素相連接的相鄰邊界上之轉接條件式及領域邊界上之邊界條件式，做數值的離散化。最後由組合定義於結構物所有的離散點處的離散化基本關係式，可得到結構物的離散方程式系統，進而求得數值解；此數值分析法除了能有系統地編成電腦程式外，因其具有較高的耦合特性，且考慮所有的基本條件，故使用較少的離散點就能得到收斂，可有效地求得精確的解，大幅降低計算機的運算量。

本篇論文應用數值積分表示微分元素法，來分析具Pasternak基座之變斷面剪變形樑之靜變形及振動問題，並且編寫求解的電腦程式，將其用於分析數個驗證例，以證明本方法分析結構物之優越性。

The differential quadrature element method (DQEM) proposed by Dr. C.N. Chen is a numerical analysis method for analyzing continuum mechanics problems. Like FEM, in using DQEM to solve a problem the domain is separated into many elements. The DQ discretization is carried out on an element-basis. The discretized governing differential or partial differential equations defined on the elements, transition conditions on inter-element boundaries and boundary conditions are assembled to obtain an overall algebraic system. The numerical procedure of this method can systematically implemented into a computer program. The coupling of solutions at discrete points is strong. In addition, all fundamental relations are considered in constructing the overall discrete algebraic system. Consequently, convergence can be assured by using less discrete points, and accurate results can be obtained by using less arithmetic operations which can reduce the computer CPU time required.

This thesis involves the application of DQEM to the deflection and vibration analyses of non-uniform Timoshenko beams on Pasternak foundation, and the related computer problem is implemented. Sample problems of static deformation and free vibration are analyzed. They prove that the developed DQEM analysis model is excellent.

摘要………………………………………………… I
英文摘要…………………………………………… II
誌謝………………………………………………… III
目錄………………………………………………… IV
表目錄……………………………………………… VI
圖目錄……………………………………………… VII
符號表…………………………………………… VIII
第一章 緒論……………………………………… 1
1-1 前言………………………………… 1
1-2 文獻回顧…………………………… 2
1-3 研究方法及目的…………………… 2
第二章 數值積分表示微分法(DQM) ……………… 3
2-1 Differential Quadrature Method(DQM)介紹……3
2-1-1 DQM的介紹…………………… 3
2-1-2 DQM和有限元素法(Finite Element Method)之不同…… 4
2-2 DQM的數學模式……………………… 5
2-3 計算權重係數的方法………………… 7
第三章 數值積分表示微分元素法(DQEM) ……… 9
3-1 DQEM的介紹………………………… 9
3-2 DQEM的求解步驟……………………… 10
3-3 權重係數之計算法………………… 11
3-3-1 多項式……………………… 11
3-3-2 以Lagrange interpolated polynomial求解………… 13
3-4 使用DQEM求解結構問題時之重要觀念…17
第四章 具Pasternak基座之變斷面剪變形樑變形問題模式……………… 18
4-1 模型建立…………………………… 18
4-2具Pasternak基座之線性變斷面剪變形懸臂樑
---自由端受集中負荷………………… 23
4-3具Pasternak基座之變斷面fix-joint剪變形樑---受均佈負荷 31
4-4具Pasternak基座之變斷面fix-fix剪變形樑---受均佈負荷… 35
4-5具Pasternak基座之兩段不等斷面剪變形懸臂樑
---自由端受集中負荷………………… 39
4-6 具Pasternak基座之兩段不等斷面剪變形懸臂樑---自由端受集中負荷(Chebyshev Polynomial不等間距節點) 43
第五章 具Pasternak基座之剪變形樑振動問題模式……………………… 47
5-1 模型建立……………………………… 47
5-2 具Pasternak基座之變斷面fix-fix剪變形樑……………… 51
5-3 具Pasternak基座之變斷面剪變形懸臂樑…………………… 53
5-4 具Pasternak基座之等斷面clamped-simply剪變形樑…… 55
5-5 具Pasternak基座之兩段不等斷面剪變形懸臂樑…………… 57
第六章 結論………………………………………… 59
參考文獻…………………………………………… 60
附錄一 五個離散點的權重係數一覽表………… A
附錄二 六個離散點的權重係數一覽表………… B
附錄三 七個離散點的權重係數一覽表…………… C
附錄四 八個離散點的權重係數一覽表………… D
附錄五 九個離散點的權重係數一覽表………… E
附錄六 十個離散點的權重係數一覽表………… F
附錄七 十一個離散點的權重係數一覽表……… G
附錄八 剪力修正係數…………………………… H

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