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研究生:吳倬民
研究生(外文):Zhuo Min Wu
論文名稱:適用於位移或速度相依之結構相依積分法
論文名稱(外文):Structure-Dependent Integration Methods for either Velocity-Dependent or Displacement-Dependent Dynamic Problems
指導教授:張順益張順益引用關係
口試委員:吳俊霖尹世洵張順益
口試日期:2018-05-24
學位類別:碩士
校院名稱:國立臺北科技大學
系所名稱:土木工程系土木與防災碩士班
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:2018
畢業學年度:106
語文別:中文
論文頁數:104
中文關鍵詞:過衝行為局部截斷誤差穩定性全外顯式積分法結構相依逐步積分法
外文關鍵詞:Over shootingLocal truncation errorStabilityFully explicit methodStructure dependent step-by-step integration method
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在進行結構動力分析時,使用逐步積分法已是普遍且有效率的方式,目前發展的趨勢為發展出擁有無條件穩定以及外顯式的簡單運算特性之逐步積分法。在發展的過程中已成功開發出半外顯式的逐步積分法,這類半外顯式積分法為無條件穩定的外顯式積分法,但在遇上具有速度相依之結構問題時還是必須進行疊代的步驟以求得速度反應。在此本文將提出一具有全外顯式數值特性的新一族逐步積分法,此新積分法結合無條件穩定及外顯式逐步積分法運算簡單的特性,並且其在計算不論是位移或速度相依的結構動力問題時都不需要使用到疊代,故此新積分法在處理任何結構問題時都可以使用外顯式的運算方式且不受穩定條件的限制,擁有這兩種特性使得新積分法可以大幅提升運算的效率。此新積分法在非線性瞬時勁度硬化時會變為有條件穩定,因此利用先前已發展的擴大穩定條件,使積分法在瞬時勁度硬化系統中也能保有無條件穩定的優勢。再將此新積分法應用在數值論例中以驗證積分法的數值特性以及實際的行為表現。
In structural dynamic, the step-by-step integration method is a universal and efficient integration method. The current development trend is to develop a step-by-step integration method that has unconditional stability and explicit formulation. A semi-explicit structure-dependent integration method has been successfully developed for time integration, where the displacement difference equation is explicit while the velocity difference equation is implicit. This implies that an iteration procedure might be still involved for velocity-dependent problems, such as that many types of viscous and viscoelastic dampers have been added into buildings to dissipate energy. A novel family of fully explicit structure-dependent integration methods is proposed for time integration. This family of integration methods can combine unconditional stability and fully explicit formulation together. Thus, it will involve no nonlinear iterations for each time step for either solving a displacement-dependent and/or velocity-dependent problems. As a result, it is very computationally efficient for solving general structural dynamics problems, where the total response is dominated by low frequency modes while high frequency responses are of no interest. In general, this family method can only have unconditional stability for linear elastic and stiffness softening systems while it will become conditionally stable for stiffness hardening systems. To overcome this difficulty, a stability amplification factor is introduced into the structure-dependent coefficients. Hence, unconditional stability for stiffness hardening systems can be also achieved. Some numerical examples are applied to confirm the numerical properties.
摘 要 i
ABSTRACT iii
誌 謝 v
目 錄 vii
表目錄 ix
圖目錄 xi
第一章 緒論 1
1.1 研究動機與目的 1
1.2 文獻回顧 3
1.3 研究內容概述 5
第二章 逐步積分法 7
2.1 積分法之收斂性 7
2.1.1 一致性 7
2.1.2 穩定性 8
2.2 新逐步積分法介紹 9
2.3 積分法之遞迴矩陣 12
第三章 數值特性 17
3.1 穩定條件 17
3.1.1 主根與頻譜半徑 17
3.1.2 穩定條件上限 18
3.1.3 擴大穩定條件 21
3.2 精確度 24
3.2.1 相對週期誤差 24
3.2.2 局部截斷誤差 25
3.3 暫態過衝行為 31
第四章 異常數值特性 47
4.1 穩態過衝行為 47
4.2 弱性不穩定 50
4.3 掌握結構非線性能力 54
第五章 數值論例 65
5.1 多自由度計算流程 65
5.2 範例一:線彈性系統 68
5.3 範例二:瞬時勁度軟化系統 69
5.4 範例三:瞬時勁度硬化系統 69
5.5 範例四:多自由度非線性彈簧系統 70
第六章 結論 99
參考文獻 101
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