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論文中文名稱:逐步積分中動態載重之離散誤差與振幅誤差之相關性 [以論文名稱查詢館藏系統]
論文英文名稱:Correlation Between Discretization Error of Dynamic Loading and Amplitude Distortion for Time Integration [以論文名稱查詢館藏系統]
院校名稱:國立臺北科技大學
學院名稱:工程學院
系所名稱:土木工程系土木與防災碩士班(碩士在職專班)
畢業學年度:103
畢業學期:第二學期
出版年度:104
中文姓名:楊尚儒
英文姓名:Shang-Ru Yang
研究生學號:102428005
學位類別:碩士
語文別:中文
口試日期:2015/06/23
指導教授中文名:張順益
指導教授英文名:Shuenn-Yih Chang
口試委員中文名:廖文義;吳俊霖
中文關鍵詞:離散化誤差振幅誤差外作用力指標參數
英文關鍵詞:discretization erroramplitude distortionexternal forceindex parameter
論文中文摘要:在逐步積分的過程中,必須選用合適的積分時間步長,方能獲得可靠的數值解。一般逐步積分法必須滿足穩定性、精確度及收斂性這三個重要的基本要求,而合適的積分時間步長則可控制由差分方程式所引起的相對週期和振幅誤差,另外,尚需抓住非線性變化及外作用力變化之特性,方能得到可靠的數值解。由先前的研究可知道積分時間步長需足夠小,以掌握外作用力變化的特性,然而到目前為止並沒有任何的指標參數來代表逐步積分法掌握外力變化之能力。本研究首先將利用結構動力學基本理論來求解單自由度結構系統受不同外力作用時的理論及數值解,並求得某一逐步積分法所引起的相對振幅誤差,將此相對振幅誤差與外作用力的相對離散化誤差之比值定義為外作用力離散化誤差所引起振幅誤差的影響參數,隨後利用極限概念求此參數之極限值,而此極限值即定義為該逐步積分法掌握外作用力變化能力之指標參數。根據此流程可針對某逐步積分法求得其指標參數,並比較探討各逐步積分法掌握外作用力的能力,以提供實際選用逐步積分法之參考。
論文英文摘要:In the step-by-step integration procedure, it is important to choose an appropriate time step to obtain a reliable solution. The three basic requirements, stability, accuracy and convergence, must be satisfied. Period distortion is often found in time integration. Hence, a reliable solution can be obtained only if the period distortion is controlled to be relatively small based on accuracy consideration. Period distortion may arise from the difference equations for displacement and velocity, linearization errors and the inaccurate representation of dynamic loading. Although it is well recognized that a step size must be small enough to capture the rapid variation of dynamic loading to yield an accurate solution, there is no criterion to determine an appropriate step size to conduct time integration. In this work, an asymptotic ratio of the relative amplitude error over the discretiztion loading error as the step size approaching zero is considered as an index number for the capability of an integration method to capture the rapid variation of dynamic loading. At first, the discretization error for each dynamic loading of sine loading, cosine loading and linear loading will be analytically evaluated. Next, the response to the abovementioned dynamic loading will be analytically obtained by using an integration method. This response will be compared to the exact response to estimate the response error. As a result, the asymptotic ratio for each dynamic loading for each integration method can be calculated. It seems that the asymptotic ratio is a simple number and a small number implies a less amplitude error.
論文目次:中文摘要 i
英文摘要 iii
誌謝 v
目錄 vii
表目錄 xi
圖目錄 xiii
第一章 緒 論 1
1.1 研究動機與目的 1
1.2 文獻回顧 2
1.3 研究內容 2
第二章 逐步積分之數值位移反應 5
2.1 理論位移反應 5
2.2 Newmark積分法之數值位移反應 7
2.3 張氏積分法之數值位移反應 11
2.4 Chen and Ricles Method之數值位移反應 15
第三章 振幅失真 19
3.1 相對振幅誤差 19
3.2 Newmark積分法之數值解誤差 19
3.2.1 正弦外作用力 20
3.2.2 餘弦外作用力 21
3.2.3 線性段外作用力 21
3.3 張氏積分法之數值解誤差 22
3.3.1 正弦外作用力 22
3.3.2 餘弦外作用力 23
3.3.3 線性段外作用力 23
3.4 Chen and Ricles Method之數值解誤差 24
3.4.1 正弦外作用力 24
3.4.2 餘弦外作用力 25
3.4.3 線性段外作用力 25
第四章 離散化誤差 37
4.1 相對離散化誤差 37
4.2 離散化誤差放大係數 40
4.3 Newmark積分法之誤差放大係數 40
4.4 張氏積分法之誤差放大係數 42
4.5 Chen and Ricles Method之誤差放大係數 43
4.6 各逐步積分法之指標參數 44
第五章 數值釋例 53
5.1 動態加載之分析假設 53
5.2 正弦外作用力之分析結果 53
5.3 餘弦外作用力之分析結果 54
5.4 線性段外作用力之分析結果 54
第六章 結論與建議 81
6.1 結論 81
6.2 建議 82
參考文獻 83
附錄 87
附錄A:Chen and Ricles Method推導 87
附錄B:張氏積分法推導 105
附錄C:Newmark積分法推導 117
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