臺灣博碩士論文加值系統

　　在土木工程的範疇裡，對結構系統的控制及保護而言，能夠即時地重建施於系統的外力，一直以來都是我們所探討且不可忽視的研究議題。

　　過去雖有許多重建外力的文獻提出，但因其運算求解費時，因此，想要達到「即時」重建的方法還尚未被廣泛研究發展應用。在本論文中，為了即時重建施加於非線性振動反算問題系統上的未知外力，我們將非線性常微分運動方程式轉換成非線性拋物型的偏微分方程式，如此一來，可以提高此法對問題抗噪之強健性。

　　隨後，我們針對外力的識別問題，利用有限差分線法，將偏微分方程 (PDE) 離散並嵌入至微分代數方程 (DAEs) 的系統中，及配合束制條件。
為了求解，我們利用增加一個虛擬時間軸的變數，將斯特姆─劉維方程式轉換為拋物型偏微分方程。

　　然後，我們可以透過隱格式李群 (implicit GL(n,R) Lie-group scheme) 法和牛頓演算法 (newton iterative scheme) ，使內外迴圈迭代增加穩定性來求解微分代數方程 (DAEs) 及求解的精度，最後，找出未知外力。這是一個很好的計算恢復力的方法，即便在較大的噪音影響下，仍有不錯的結果。

　　從表面上來看，我們似乎將一個簡單的常微分方程 (ODE) 轉換成一個更複雜的偏微分方程 (PDE) ，但是當我們在計算一些非線性的振動反算問題時，不管是在長時間計算下或是在較大噪音的影響下，我們仍可以得到好的結果。我們可以減輕噪音的影響，是因為透過有限差分線法，我們將原本的偏微分方程，離散成2m條常微分方程式，而會受到噪音的干擾只有第一條和最後一條方程式，中間的常微分方程式，可以平均分擔其影響，因此，透過這個偏微分方程的轉換，我們可以將噪音的影響縮減到最小。

　　隨著線性結構、杜芬方程式、范德波爾方程式、杜芬─范德波爾方程式及座椅系統等五個數值算例顯示，此方法呈現相當好的精度與效率，且簡單易於使用，未來發展應用的機會極大。

　　For structures protection and control, it is utmost to immediately detect the external force being imposed on the structures currently in civil engineering.

　　Desiring a real-time recovery of an unknown external force in the nonlinear inverse vibration problem; in this thesis, we transform the nonlinear ordinary differential equation (ODE) of motion into a nonlinear parabolic type partial differential equation (PDE), which can raise the robustness against large noise. Then we come to an unknown external force identification problem, of which the numerical method of lines is used to discretize the governing equation into a system of differential algebraic equations (DAEs) and with the constraints conditions.

　　A fictitious time variable transforming the Sturm-Liouville equation into a parabolic type partial differential equation (PDE) endowed with an extra fictitious time dimension.

　　Hence, we can develop an implicit GL(n, ) Lie-group scheme and a Newton algorithm to stably solve the DAEs for finding the unknown force, damping function, or stiffness function, which is well recovered even under a large noise.

　　Obviously, we transform a simple ODE into a more complex PDE; however, the merits obtained in this transformation will be seen, when we examine some nonlinear inverse vibration problems with large time span and under large noise. We can alleviate the influence of noise, which is only happened at the first and the last line equations among the many 2m equations.The estimated results obtained by the novel methods are quite promising.

誌謝 I
摘要 III
ABSTRACT V
目錄 VII
圖目錄 IX
第一章 緒論 1
1.1 前言 1
1.2 文獻回顧 3
1.3 論文架構 5
第二章 理論背景與數值分析方法 7
2.1 群 7
2.1.1 群的歷史 8
2.1.2 群的定義 9
2.1.3 李群與李代數 10
2.2 數值分析方法 12
2.2.1 微分方程中的 結構 14
2.2.2 隱格式李群 法 16
2.2.3 牛頓法求解非線性代數方程 20
2.2.4 數值計算流程圖 21
第三章 化為偏微分方程的李群微分代數即時估測方法 23
3.1 ODE轉換成 PDE 23
3.2 有限差分線法 26
3.3 李群微分代數方程（LGDAE）法 28
3.4 隱格式李群 30
3.5 數值算法 33
第四章 數值算例 37
4.1 算例一 Linear 37
4.1.1 計算量測位移無噪音影響下之外力 38
4.1.2 計算當量測位移具有噪音影響下之外力 40
4.2 算例二 Duffing oscillator 44
4.2.1 計算量測位移無噪音影響下之外力 45
4.2.2 計算當量測位移具有噪音影響下之外力 48
4.3 算例三 Van der Pol Oscillator 58
4.3.1 計算量測位移無噪音影響下之外力 59
4.3.2 計算當量測位移具有噪音影響下之外力 61
4.4 算例四 Duffing-Van der Pol Oscillator 65
4.4.1 計算量測位移無噪音影響下之外力 66
4.4.2 計算當量測位移具有噪音影響下之外力 68
4.5 算例五 座椅 73
4.5.1 計算量測位移無噪音影響下之外力 75
4.5.2 計算當量測位移具有噪音影響下之外力 78
第五章 結語與未來展望 87
5.1 結語 87
5.2 未來展望 89
參考文獻 90

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