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研究生:葉祖銘
研究生(外文):Zu-Ming Ye
論文名稱:杜芬方程式具分數微分阻尼之動力分析
論文名稱(外文):Dynamic Analysis of the Fractionally-damped Duffing Equation
指導教授:許隆結陳献庚
指導教授(外文):Long-Jye SheuHsien-Keng Chen
學位類別:碩士
校院名稱:中華大學
系所名稱:機械與航太工程研究所
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2006
畢業學年度:94
語文別:中文
論文頁數:38
中文關鍵詞:分數微分阻尼渾沌杜芬方程式
外文關鍵詞:Fractionally dampedChaosDuffing equation
相關次數:
  • 被引用被引用:1
  • 點閱點閱:149
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  • 下載下載:0
  • 收藏至我的研究室書目清單書目收藏:0
分數微分及其應用在過去二、三十年已有驚人的進展。而且最近幾年分數微分阻尼系統的振動現象也已經有許多學者投入其研究。本論文係研究具分數微分阻尼的杜芬方程式受周期外力下的響應,並探討分數微度的阻尼對系統動力行為的影響,以數值模擬作ㄧ完整的動力分析。在數值分析中,本文以Caputo所提出的分數微分定義將系統方程先轉換成分數積分方程組,並以Adams-Bashforth-Moulton Predictor-Corrector方法求解此方程組。在研究中以上述的數值方法利用分岐圖分析在不同微度α下系統的動力行為。此外,本文藉由繪出此非線性系統的相軌跡圖、龐茄萊映射圖來探討系統的規則與渾沌行為,並以李雅普諾夫指數來確認系統是否為渾沌行為。文中並分析在週期外力的振幅f與頻率 的改變對於系統的影響,利用α-f及α- 參數平面表做完整的動力行為分析。研究結果發現在多個參數下系統產生混沌運動,並發現系統經由倍週期路徑而達到渾沌行為之現象。
The vibrations of the fractionally damped systems have attracted increasing attentions in recent years. The dynamics of the fractionally damped Duffing equation is examined in this study. The fractionally damped Duffing equation is transformed into a set of fractional integral equations which are solved by a predictor-corrector method. The effect of fractional order of damping on the dynamic behaviors of the motion is mainly studied. Bifurcation of the parameter dependent system is drawn numerically. The time evolutions of the nonlinear dynamical system responses are described in phase portraits and the Poincaré map technique. The occurrence and the nature of chaotic attractors are verified by evaluating the largest Lyapunov exponents. Moreover,the effect of amplitude and frequency of external force on the dynamic behaviors of the motion is also analyzed.Period doubling routes to chaos are also found in this study.
摘 要 i
Abstract ii
誌 謝 iii
目 錄 iv
圖 目 錄 vi
表 目 錄 viii
符號說明 ix
第一章 緒論 1
1.1 研究動機 1
1.2 研究目的 4
1.3 文獻回顧 4
1.4 研究方法 5
第二章 數學理論與數值方法 7
2.1 問題描述與分數微分定義 7
2.2 數值方法 9
第三章 結果與討論 11
3.1 固定參數下系統在不同分數微度之動力行為分析 11
3.2 週期外力之振幅f與頻率ω對系統動力行為之影響 14
第四章 結論 33
第五章 未來展望 34
參考文獻 35
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