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研究生:張育誠
研究生(外文):Yu-Cheng Chang
論文名稱:多自由度杜芬微分方程組的保能及保群計算方法
論文名稱(外文):The energy and group preserving schemes for multi degree of freedoms Duffing equations
指導教授:劉進賢
口試委員:陳永為郭仲倫
口試日期:2015-06-30
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:土木工程學研究所
學門:工程學門
學類:土木工程學類
論文種類:學術論文
論文出版年:2015
畢業學年度:103
語文別:中文
論文頁數:120
中文關鍵詞:杜芬微分方程式保能算法保群算法四階龍格-庫塔方法
外文關鍵詞:Duffing equationenergy preserving schemes (EPS)group preserving schemes(GPS)4th order Runge-Kutta (RK4) method
相關次數:
  • 被引用被引用:0
  • 點閱點閱:101
  • 評分評分:
  • 下載下載:0
  • 收藏至我的研究室書目清單書目收藏:0
在工程與數學應用中,非線性振動是個相當常見的問題,過去的文獻中已經有許多方法可以求解非線性振動的問題,但它們往往都忽略了能量守恆這個議題。在本篇論文中,我們提出了保能算法(EPS),將無阻尼與無外力情況下的杜芬微分方程組透過李群轉換成常微分方程組,求解過程中保能算法能夠自動保持能量守恆,使得能量在長時間計算維持不變;接下來我們會繼續探討加上阻尼與外力的杜芬微分方程組,這個部分我們將使用有效且具有高精確度的保群算法(GPS)求解。最後,我們會將整個問題延伸到二維空間與三維空間中,同樣地,我們可以使用保能算法與保群算法去求解二維與三維的杜芬微分方程組。此外,我們將使用四階龍格-庫塔(RK4)方法與保能算法以及保群算法做比較,因為四階龍格-庫塔方法能夠有效地求解微分方程問題,且同時具有四階的精度,所以求出的結果相當值得我們信賴。藉由與四階龍格-庫塔方法求得之解做比較,我們可以得知EPS與GPS的優點、精確性,當然還可以藉由比較每一步所產生的能量誤差得知EPS的保能效果與優越性。

In engineering and mathematics fields, the oscillatory problems of nonlinear oscillators are common problems. There are many computational methods which have been developed for solving the nonlinear oscillatory problems. However, most of these methods can not retain the energy. In this thesis, we develop a novel energy preserving scheme (EPS) for the undamped and unforced Duffing equation by recasting it to a Lie-type ordinary differential equation. The EPS can automatically preserve the total energy to be a constant value in a long term computation. Then, we will extend this problem to the damped and forced Duffing equations. Here, we use the group preserving schemes (GPS) to solve the problems, which can solve the problems effectively and accurately. Finally, we extend the problems to the coupled Duffing equations and three degrees of freedom Duffing equations. Also, we still can use the EPS and the GPS to solve the problems accurately. In each problem, we also compare the present results with the solution obtained by the fourth order Runge-Kutta (RK4) method, which has fourth-order accuracy. By comparing the EPS and RK4, we can see the advantages, accuracy and capability of preserving energy of the EPS.

口試委員審定書 i
誌謝 ii
摘要 iii
ABSTRACT iv
目錄 v
圖目錄 viii
第一章 緒論 11
1.1 前言 11
1.2 文獻回顧 12
1.3 研究動機與目的 13
1.4 論文架構 13
第二章 理論基礎 15
2.1 群 15
2.2 李群 16
2.3 光錐構造 18
2.4 李代數 20
2.5 增廣動態系統 22
2.6 凱萊轉換(Cayley Transform) 26
2.7 指數映射(Exponential Mapping) 31
2.8 數值積分方法 - 尤拉法 35
2.9 數值積分方法 - 龍格-庫塔法 35
2.10 杜芬微分方程式 36
第三章 多自由度杜芬微分方程組 38
3.1 一維保能算法 38
3.1.1 非線性彈簧勁度係數β>0 38
3.1.2 非線性彈簧勁度係數β<0 40
3.2 一維保群算法 41
3.2.1 非線性彈簧勁度係數β>0 42
3.2.2 非線性彈簧勁度係數β<0 44
3.3 二維保能算法 47
3.3.1 非線性彈簧勁度係數β1>0與 β2>0 48
3.3.2 非線性彈簧勁度係數β1>0與 β2<0 53
3.3.3 非線性彈簧勁度係數β1<0與 β2>0 58
3.3.4 非線性彈簧勁度係數β1<0與β2<0 63
3.4 二維保群算法 68
3.4.1 非線性彈簧勁度係數β1>0與β2>0 69
3.4.2 非線性彈簧勁度係數β1>0與β2<0 71
3.4.3 非線性彈簧勁度係數β1<0與β2>0 73
3.4.4 非線性彈簧勁度係數β1<0與β2<0 76
3.5 三維保能算法 78
3.6 三維保群算法 83
第四章 數值算例 87
4.1 數值算例一 87
4.2 數值算例二 88
4.3 數值算例三 89
4.4 數值算例四 90
4.5 數值算例五 90
4.6 數值算例六 91
4.7 算例圖 93
第五章 結論與未來工作 117
參考文獻 119


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