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論文中文名稱:不具數值阻尼之結構相依積分法的比較研究 [以論文名稱查詢館藏系統]
論文英文名稱:Comparative Study of Non-Dissipative Structure-Dependent Integration Methods [以論文名稱查詢館藏系統]
院校名稱: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
畢業學年度:106
畢業學期:第二學期
出版年度:107
中文姓名:陳彥蓉
英文姓名:Yen-Rong Chen
研究生學號:105428008
學位類別:碩士
語文別:中文
口試日期:2018/05/24
論文頁數:119
指導教授中文名:張順益
口試委員中文名:張順益;尹世洵;吳俊霖
中文關鍵詞:結構相依逐步積分法局部截斷誤差過衝行為擴大穩定條件弱性不穩定
英文關鍵詞:Structure dependent step-by-step integration methodLocal truncation errorOver shootingStabilityWeak instability
論文中文摘要:本文所探討的四種結構相依逐步積分法其位移與速度差分方程式的係數不同於一般積分法所使用的純量常數,而是以質量、阻尼與勁度等矩陣所組合而成的係數矩陣,而因積分法都具有結構相依的性質,使得這些積分法在線性以及軟化系統的情況下具有無條件的穩定性,然而在硬化系統時則會受到穩定條件的限制,因此本文利用過去研究已發展的擴大穩定條件的方法將其進行探討與改進。雖然四種結構相依積分法在發展初期已針對於基本數值特性進行完整的探討,然而對於非線性的問題則未深入探討。另外在實際應用時也出現了異常的數值特性,例如穩態過衝行為、弱性不穩定與掌握高度非線性能力等,因此本篇論文的重點在於對各積分法進行非線性的數值特性分析,並以非線性的數值論例模擬實際的結構行為,以檢驗各積分法在進行實際的結構工程應用時的表現。分析結果顯示出各積分法皆展現出二階精確度的無條件穩定,各積分法皆不具有暫態過衝,在穩態反應中則產生出過衝行為,透過加入載重相依項 ,可將強迫振動的局部截斷誤差中不利的誤差項去除,則將各積分法的穩態過衝行為修正。對於弱性不穩定方面,TLM與CFM2積分法則具有弱性不穩定,在掌握高度非線性能力方面,CEM與CFM2積分法展現較差的能力。經由本論文的分析結果,在四種結構相依的外顯式積分法中,CFM1積分法在各項數值特性中展現出最優良的行為,其不具有暫態過衝行為以及弱性不穩定,在修正項加入後可以將穩態過衝行為消除,並且CFM1積分法具有掌握高度非線性之能力,因此本論文最後推薦CFM1積分法作為結構動力分析中最具實際應用價值之積分法。
論文英文摘要:Four structure-dependent integration methods are explored and compared in this work. These integration methods are different from conventional integration methods since their coefficients of the difference equations are not scalar constants but can be functions of the product of the initial structural properties and step size. Either favorable or adverse numerical properties of each structure-dependent integration method are thoroughly explored. For comparison purpose, numerical properties of the commonly used Newmark family method are also investigated. The favorable properties include unconditional stability, second-order accuracy, explicit formulation and no overshoot in both transient and steady-state responses. Whereas, the adverse properties that might be experienced for structure-dependent integration methods are conditional stability for stiffness hardening systems, a high frequency overshoot in steady-state responses, a poor capability of capturing structural nonlinearity and a weak instability.
All the four structure-dependent integration methods can only have conditional stability for stiffness hardening systems. Hence, a stability amplification factor can be applied to enlarge the unconditional stability interval. As a result, an unconditional stability can be achieved for certain stiffness systems after using an appropriate stability amplification factor except for TLM. In addition, these integration methods also show a high frequency overshoot in steady-state responses. However, they can be eliminated by introducing a load-dependent term into the displacement difference equation. It is very important to find that CRM and TLM have a weak instability property, which may lead to an inaccurate solution or even numerical instability. Although CEM and CFM2 generally have no weak instability, they exhibit a poor capability for seizing structural nonlinearity. Notice that CRM is a special member of CFM2 and is the only member that exhibits a weak instability for CFM2. In general, CFM1 can have all the desired numerical properties while it does not show any above-mentioned adverse properties. Consequently, it is strongly recommended for practical applications.
論文目次: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
2.1 逐步積分法介紹 7
2.1.1 NFM 7
2.1.2 CFM1 8
2.1.3 CEM 8
2.1.4 TLM 9
2.1.5 CFM2 9
2.2 積分法之遞迴矩陣 10
2.3 積分法之收斂性 15
2.3.1 一致性 15
2.3.2 穩定性 17
第三章 數值特性 21
3.1 穩定條件 21
3.1.1 主根與頻譜半徑 21
3.1.2 穩定條件上限 23
3.1.3 擴大穩定條件 28
3.2 精確度 31
3.2.1 相對週期誤差 31
3.2.2 局部截斷誤差 33
3.3 暫態過衝行為 37
第四章 異常數值特性 73
4.1 穩態過衝行為 73
4.2 弱性不穩定 75
4.3 掌握高度結構非線性能力 84
第五章 數值論例 95
5.1 多自由度計算流程 95
5.1.1 NFM 95
5.1.2 CFM1 96
5.1.3 CEM 97
5.1.4 TLM 98
5.1.5 CFM2 99
5.2 範例一:線彈性系統 100
5.3 範例二:瞬時勁度軟化系統 100
5.4 範例三:瞬時勁度硬化系統 101
5.5 範例四:多自由度非線性彈簧系統 101
第六章 結論 115
參考文獻 117
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論文全文使用權限:同意授權於2018-06-01起公開