修正結構動態再分析之探討

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王萬益(Wan-yi Wang) 查詢紙本館藏

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機械工程學系

論文名稱

修正結構動態再分析之探討
(Eigenvalue Reanalysis of Structural Modification)

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摘要(中)

結構修正再分析問題對於結構分析、再設計及最佳化過程中不斷扮演關鍵性角色。本文在對已有的一些研究文獻和成果綜述的基礎上，就如何提高結構修正動態再分析方法的精度、計算速度以及收斂範圍等內容開展了進一步研究。
隨著科技的進步，人們對工程結構的特性和品質要求越來越高，很多複雜因素在設計階段就必須加以考慮，導致結構動力學模型自由度數目的日益增長，應用正規方法對其動態特性進行一次計算分析就需耗費可觀的時間、經費和人力，有時甚至是行不通的。故如何降低計算花費是結構修正再分析的重點。結論是要降低計算花費，必須在進行修正再分析時，避免對修正結構再進行完整分析，經前人之研究，以初始結構之動態資訊作為再分析之依據，選擇適當的再分析方法，將可有效達到預期目標。因此初始結構動態資訊之精確與否，將影響再分析之結果。本文以矩陣為基礎之Chebyshev 譜法，即所謂的微分化矩陣，解析結構動態特性。並經由Matlab矩陣特徵值指令去分析具有不同邊界條件之結構的模態參數。接著，利用初始結構資訊進行結構再分析探討。首先由精確解為著眼點，以代數方法探討修正結構再分析問題，以等慣性轉換求解動態勁度矩陣之隱根，並導出將特徵值定位的計算方法；繼而在隱根為已知下探討隱向量的特質及解法。接著探討修正結構再分析之近似解法。本文舉出多種近似解法並以數值例比較各算法之精確度，進而選用以縮減基為依據之組合近似法作為再分析最佳工具。作者首先探討縮減基之理論背景，進而以此為依據深入綜整組合近似法於靜態結構及動態結構之解法及其精確度之掌握。其結果顯示無論結構小修正或大修改，組合近似法均能有效且精確的得到近似解。

摘要(英)

The problem of structural modification and reanalysis plays a key role in the structure analysis, redesign and optimization of a structural system. Based on the previous researches, the present thesis tends to investigate deeply the methods of structural modification and dynamic reanalysis to increase the accuracy and computational efficiency under the same convergent criterion.
As the advance of technology, the request for the characteristics and qualification of engineering structure becomes higher and higher. Many complex factors must be taken into consideration during the design stage, and it results in the increase of freedom in the structural dynamic model. The conventional, one-time computational analysis for the dynamic characteristics is highly consumed in time, cost, and man-power, and may not be workable sometime. The structural modification and reanalysis aims to reduce the computational cost. To reduce the computational cost, a complete analysis of the modified structure is avoided in the redesign and reanalysis stage. Based on the previous researches, the dynamic information of the original structure can be used as the basis of the reanalysis. A proper analytical method, plus accurate dynamic information of the original structure, will largely affect the results of reanalysis.
The present thesis analyzes the structural dynamic characteristics by the matrix-based Chebyshev spectral method, or so called differentiation matrix method. A m-file of matrix characteristic value in Matlab is then adopted to analyze the model parameters of the structure subjected to different boundary conditions. The structure reanalysis is then preceded based on the initial structural information.
To assure accurate solutions, algebraic method is used to investigate the modified structural, re-analysis problem, equal inertia transformation is then used to find out the implicit roots of the dynamic, stiffness matrix. A computational method is derived to find out the eigenvalues. Based on the known implicit roots, the characteristics and solutions of the implicit vectors are investigated. The present thesis illustrates several approximate methods and compares the corresponding accuracy. A combined approximation method based on the reduced basis is then used as the best tool of reanalysis. The present thesis investigates the theoretical ground of reduced basis, and then derives the solutions by combined approximate method for static structural system, as well as dynamic structural system. Results show that the combined approximate can effectively and accurately find out the approximate solutions for a minor modified structure, even for a major modified structure.

關鍵字(中)

★ 特徵值

關鍵字(英)

★ Eigenvalue

論文目次

謝誌..............................................................................................................I
摘要.............................................................................................................II
ABSTRACT.........................................................................................................IV
目錄.............................................................................................................VI
附圖目錄.........................................................................................................XI
附表目錄........................................................................................................XII
符號說明........................................................................................................XIV
第一章導論......................................................................................................1
1.1 結構動態再分析技術的發展與現狀.............................................................................1
1.2 結構修正後動態再分析的主要分類.............................................................................3
1.3 本文架構...................................................................................................5
第二章以矩陣為基礎之契巴雪夫譜法解析結構動態特性................................................................8
2.1 微分化矩陣及Chebyshev 譜方法............................................................................10
2.1.1 微分化矩陣(differentiation matrix).....................................................................10
2.1.2 Chebyshev 譜方法.......................................................................................12
2.2. 邊界值(boundary value)問題...............................................................................14
2.3.邊界條件於Chebyshev 譜矩陣中之處理........................................................................17
2.4. 例題:以Chebyshev 譜法解析非均勻Timo shenko樑之自由振動
..........................................................................................................19
2.5.數值例結果................................................................................................24
2.6.結論......................................................................................................26
第三章以代數方法探討修正結構再分析問題之精確解.................................................................31
3.1結構修改之統御方程式........................................................................................32
3.2修改後結構之模式分析.......................................................................................34
3.3矩陣之譜切(Spectrum Slicing)公式...........................................................................35
3.4數值勘根(Root-finding)法...................................................................................37
3.5求解隱向量.................................................................................................40
3.6數值例與討論...............................................................................................43
3.7結論.......................................................................................................45
第四章修正後結構動態再分析近似解之方法.........................................................................49
4.1.矩陣微擾之特徵值分析......................................................................................50
4.2改良型矩陣微擾法...........................................................................................52
4.2.1 William B. Bickford法......................................................................................52
4.2.2 Chen’s法..................................................................................................53
4.3.混合法.......................................................................................................53
4.3.1 Pade’ 近似法..............................................................................................53
4.3.2.組合近似法.................................................................................................55
4.5.數值例.......................................................................................................55
第五章以縮減基(reduced basis)理論探討結構再分析................................................................58
5.1. 縮減基之應用基礎.........................................................................................59
5.2 一維設計變數局部近似之推導................................................................................59
5.3 多維設計變數局部近似之推導................................................................................63
5.4. 導數模式之實際計算.......................................................................................64
5.5. Krylov 序列與 Krylov 次空間..............................................................................65
5.6 十段組合椼樑之導數模式....................................................................................66
5.7. Krylov 次空間法與組合近似法之等效關係....................................................................67
第六章組合近似法於結構再分析之應用.............................................................................73
6.1 靜態結構組合近似法再分析..................................................................................73
6.1.1修正結構靜態再分析架構...................................................................................73
6.1.2. 組合近似法.............................................................................................75
6.1.2.1以縮減基描述位移向量...................................................................................75
6.1.2.2 選擇基向量............................................................................................75
6.1.2.3 建立獨立基向量........................................................................................76
6.1.2.4 數值例................................................................................................78
6.2. 組合近似法在動態結構再分析之應用.........................................................................80
6.2.1結構動態分析.............................................................................................80
6.2.2 修正結構動態再分析......................................................................................80
6.2.3 特徵值問題之逆疊代(Inverse Iteration)解法...............................................................81
6.2.4 組合近似法之特徵問題再分析.................................................................................81
6.2.5 基向量之確認............................................................................................82
6.2.6改良振態模式.............................................................................................83
6.2.7基向量之Gram-Schmidt 正交化..............................................................................84
6.2.8基向量之飄移(shift)......................................................................................85
6.2.9 具飄移逆疊代之組合近似法................................................................................86
6.2.10 數值例.................................................................................................87
6.3以組合近似法求解動態結構靈敏度.............................................................................91
6.3.1. 靜態結構之靈敏度.......................................................................................92
6.3.2 組合近似法計算修正位移導數.............................................................................93
6.3.2.1 計算靜態修正結構位移..................................................................................93
6.3.2.2 靜態修正結構位移一次導數..............................................................................94
6.3.2.3 靜態修正結構位移之二次導數............................................................................96
6.3.3. 動態結構之靈敏度.......................................................................................96
6.3.4 數值例..................................................................................................98
6.4 結論....................................................................................................100
第七章結論与展望..............................................................................................103
7.l 結論.....................................................................................................103
7.2未來研究方向..............................................................................................105
參考資料........................................................................................................107
附錄............................................................................................................117

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指導教授

洪勵吾(Lih-wu Hourng)

審核日期

2011-7-27

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