臺灣博碩士論文加值系統

本研究之目的為應用近代數值方法，Laplace Adomian分解法(LADM)來求解非線性樑之自由振動與大撓度變形的問題，找出不同情況下對樑之振動頻率或大撓度變形之影響。第一部分應用LADM將Euler-Bernoulli樑之統御方程式轉換成迭代方程，利用遞迴關係式得到近似解析解，接著對近似解進行簡單代數計算，求出模型的自然頻率與振動模態。透過改變模型的參數來探討其對動態系統之自然頻率的影響，參數包含幾何形狀、移動及旋轉彈簧模數、端點質塊重量、偏心距離及軸向力等。第二部分則利用LADM來探討Euler-Bernoulli樑在端點受不同大小的外力及彎矩的作用下，樑之大撓度變形情形。
研究結果得知，自然頻率隨會著移動彈簧模數、旋轉彈簧模數、軸向張力強度的增加而提升；當幾何差異越明顯時，其各項自然頻率會越接近。考慮大撓度變形的情況下，彎矩的作用相對於力的作用更能對樑產生形變，且其變形是越明顯易見的。本文中所得之計算結果，亦與相關文獻或解析結果相吻合，故LADM是一種精確又簡明的數值方法。

In this study, the Laplace Adomian decomposition method (LADM) was used to analyze free vibration of a nonlinear beam and large deflection of a cantilever beam. The relationship between structure parameters and natural frequencies or deflection was also figured out.
In the first section, the characteristic/eigenvalue equation and mode shape functions of a general beam were analytically derived by LADM. After that, effects of different physical parameters including geometry formula, translational or rotational spring constant, magnitude of axial tensile force, and the eccentricity of the tip mass on natural frequencies were investigated. In the second part, the governing equation of the large deflection was carried out with Euler–Bernoulli moment–curvature relationship. Next, the deflection under non-following end force and end moment was probed with LADM through this part.
The results of this study indicated that natural frequencies of the beam would increase with higher translational or rotational spring constant and magnitude of axial tensile force. On the other hand, the first natural frequency would decrease, and the other frequencies would increase when the eccentricity of the tip mass was larger. As for the structure geometry, the more complex it was, the closer the natural frequencies would they be. For large deflection cases, the results revealed that the influence of end moment was more obvious than the influence of end force. Further, end moment would cause obvious deflection near the tip. The end force, by contrast, would cause the deflection through the whole beam.

摘要 I
Extended Abstract II
致謝 XI
目錄 XII
表目錄 XV
圖目錄 XVII
符號說明 XIX
第一章 緒論 1
1-1 研究動機與目的 1
1-2 文獻回顧 3
1-2-1 樑之振動問題研究 3
1-2-2 Adomian分解法 4
1-3 本文架構 6
第二章 Laplace Adomian分解法 (LADM) 7
2-1 Adomian分解法 (ADM) 8
2-2 Adomian多項式 (Adomian polynomial) 10
2-2-1 非線性多項式 (Nonlinear Polynomials) 11
2-2-2 三角函數 (Trigonometric) 13
2-3 修正Adomian分解法 (MADM) 16
2-3-1 修正ADM (一) 16
2-3-2 修正ADM (二) 21
2-4 Laplace Adomian分解法 (LADM) 25
2-4-1 運算法則 25
2-4-2 LADM在特徵值問題之應用 32
2-4-3 Padé近似法 39
2-4-4 結論 41
第三章 非均勻Euler-Bernoulli樑之自由振動分析 45
3-1 模型建立 45
3-2 均勻Euler-Bernoulli樑 49
3-2-1 統御方程式與邊界條件 49
3-2-2 LADM求解均勻樑之自然頻率 50
3-3 非均勻Euler-Bernoulli樑 65
3-4 結論 80
第四章 Euler-Bernoulli樑之大撓度變形分析 93
4-1 模型建立 93
4-2 LADM解題流程 95
4-3 數值結果與討論 97
第五章 總結與建議 102
5-1 總結 102
5-2 未來研究方向與建議 104
參考文獻 105

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