楊永斌臺灣大學:土木工程學研究所陳俊德Chen, Chun-TehChun-TehChen2010-07-012018-07-092010-07-012018-07-092009U0001-1708200909432800http://ntur.lib.ntu.edu.tw//handle/246246/187904傅立葉轉換(Fourier Transformation)之應用有諸多限制條件，無法完全適用於非線性系統。為了能分析非穩態(non-stationary)或非線性過程(nonlinear processes)之訊號，本文採用具有較高適用性之希爾伯特–黃轉換(Hilbert-Huang Transform–HHT)作為分析工具之一。 希爾伯特–黃轉換主要包含兩部分之處裡流程：(1)經驗模態分離法(Empirical Mode Decomposition–EMD)：可將訊號分離成數個內建模態函數(Intrinsic Mode Function–IMF)，而每個IMF皆具有良好的希爾柏特轉換特性。(2)希爾柏特轉換(Hilbert Transform)：可得到訊號之即時頻率與即時振幅，若繪製成能量–頻率–時間分佈圖，則稱為希爾伯特頻譜(Hilbert Spectrum)。 本文以雙桿桁架系統作為測試模型，考慮幾何非線性效應，利用有限元素法配合Newmark法進行數值分析，並提供對應之達芬方程式參數。本文嘗試由頻率的角度，研究非線性系統之動力行為，比較FFT與HHT分析結果之差別，並針對振動頻率的變化、週期倍增以及混沌現象等，進行較為系統的探討。In the Fourier analysis, the fundamental assumption of linear and stationary process is required for the data. Applying the Fourier analysis to those data generated from nonlinear systems may cause misunderstanding of the physical phenomena hidden in the data. On the other hand, the Hilbert-Huang transform (HHT) is considered more suitable for analyzing nonlinear and non-stationary data. HHT includes two major parts: (1) empirical mode decomposition (EMD): a sifting process by which the data can be decomposed into a collection of intrinsic mode functions (IMF) that admit well-behaved Hilbert transforms; (2) Hilbert transform: a type of transform by which the instantaneous frequency and amplitude can be calculated for any instant. The energy distribution being plotted in a 3-D energy-frequency-time space is designated as the Hilbert spectrum. A two-member truss system with the effect of geometric nonlinearity considered is taken as the example in this study. The dynamic response of such a system is numerically analyzed by the finite element method along with the Newmark method, with the corresponding parameters in the Duffing equation given in each case. By comparing the results obtained from both the FFT and HHT analyses in frequency domain, the dynamic behavior of the nonlinear system is systematically studied, especially with respect to the variation in frequency caused by the geometric nonlinearity, period-doubling, chaos phenomenon, and so on.誌 謝 ......................................... I 要 ......................................... II文摘要....................................... III一章 導論.1 研究動機與目的 ............................ 1.2 研究範圍 .................................. 2二章 結構非線性增量理論.1 非線性推演法簡介 .......................... 3.2 參考狀態說明 .............................. 3.3 虛功方程式推導 ............................ 4.4 更新式Lagrange推演法 ...................... 5.5 剛體運動法則 .............................. 8.6 結論 ...................................... 8三章 有限元素增量分析.1 有限元素增量平衡方程式推導 ................ 11.1.1 二維桁架元素之勁度矩陣 .................. 11.1.2 二維桁架元素之質量矩陣 .................. 16.1.3 二維桁架元素之阻尼矩陣 .................. 17.2 二維桁架元素之剛體測試 .................... 17.3 廣義位移控制法 ............................ 18.4 有限元素非線性靜力分析流程 ................ 22.4.1 預測階段與校正階段 ...................... 22.4.2 增量–迭代分析流程 ...................... 23.5 有限元素非線性動力分析流程 ................ 25.5.1 預測階段與校正階段 ...................... 26.5.2 增量–迭代分析流程 ...................... 28.6 結論 ...................................... 30四章 希爾伯特–黃轉換之基本理論.1 前言 ...................................... 32.2 希爾伯特轉換 .............................. 32.3 經驗模態分離法 ............................ 34.4 曲率篩選法 ................................ 35五章 桁架系統動力分析.1 前言 ...................................... 43.2 雙桿桁架系統介紹 .......................... 43.3 分析結果說明 .............................. 45.3.1 無阻尼之自由振動系統 .................... 45.3.2 含阻尼之自由振動系統 .................... 48.3.3 含阻尼之強迫振動系統 .................... 49六章 結論與未來展望.1 結論 ...................................... 82.2 未來展望 .................................. 83考文獻 ...................................... 842925125 bytesapplication/pdfen-US桁架非線性動力混沌希爾柏特轉換trussnonlineardynamicchaosHilbert TransformFFTHHTthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/187904/1/ntu-98-R95521224-1.pdf