本文主要是探討樑受初始負載下之自由振動行為，以彈性力學之平面應變為基礎，利用有限元素法推導出自由振動方程式，與一般傳統樑理論所做的分析有所差異，能夠更準確的求取樑結構受力下之自然振動頻率。在不同邊界條件及幾何比例下，本文討論受軸向負載的樑之自然振動頻率，並將結果與Timoshenko樑理論做比較。另外當樑承受均勻縱向負載時，分別考慮其所造成的初始應力以及同時考慮初始應力及初始位移時，分析其振動頻率，以了解初始應力與初始位移在預加負載樑結構振動分析上的影響。

In this thesis, free vibration of beam structures with initial load is analyzed. It is based on the theory of elasticity and the formulation is derived by finite element method. The present method is different from traditions beam theories and is able to simulate beam vibration more precisely. The free vibration frequencies of beams under initial axial loading and different boundary conditions and geometrical aspect ratios are solved first and compared with those of Timoshenko beam theory. Vibration of beams with initial lateral loading is then analyzed, considering only the initial stresses and both the initial stresses and the initial displacements induced by the initial loading, to reveal the effect of initial stresses and initial displacement on vibration of beams with initial loading.

摘要 …………………………………………………………………i
目錄 ………………………………………………………………iii
表目錄 ………………………………………………………………v
圖目錄 ………………………………………………………………vii
第一章 緒論 ……………………………………………………1
1-1 前言 ……………………………………………………1
1-2 文獻回顧 …………………………………………………2
1-2-1 尤拉樑理論 ……………………………………2
1-2-2 Timoshenko樑理論 ……………………………5
1-2-3 高階剪切變形理論 ……………………………8
1-2-4 三維彈性力學分析法 …………………………10
第二章 樑受預應力之平面應變有限元素振動分析 ……………12
2-1 前言 ……………………………………………………12
2-2 理論推導 …………………………………………………12
2-2-1 自由振動方程式 …………………………………12
2-2-2 樑的初始應力 …………………………………15
2-2-3 樑的初始位移 …………………………………18
第三章 問題解析 …………………………………………………19
3-1 前言 ………………………………………………………19
3-2 問題描述 …………………………………………………19
第四章 結果討論 …………………………………………………23
4-1 前言 ………………………………………………………23
4-2 軸向負載樑振動分析 ……………………………………23
4-3 縱向負載樑振動分析 ……………………………………23
第五章 結論 ………………………………………………………45
參考文獻 …………………………………………………………46
附錄 ……………………………………………………………50
表4-1: 本文單獨考慮初始應力與Timoshenko樑理論之自然振動比較 ……………………………………………………………………26
表4-2: 本文單獨考慮初始應力與Timoshenko樑理論之自然振動比較 ……………………………………………………………………26
表4-3:樑結構受初始縱向均勻負載時之無因次化自然振動頻率初始應力P_P ………………………………………………………………27
表4-4:樑結構受初始縱向均勻負載時之無因次化自然振動頻率初始應力C_C ………………………………………………………………29
表4-5:樑結構受初始縱向均勻負載時之無因次化自然振動頻率初始應力C_F ………………………………………………………………31
表4-6:樑結構受初始縱向均勻負載時之無因次化自然振動頻率初始應力C_P ………………………………………………………………33
表4-7:樑結構受初始縱向均勻負載時之無因次化自然振動頻率初始應力+初始位移P_P ……………………………………………………35
表4-8:樑結構受初始縱向均勻負載時之無因次化自然振動頻率初始應力+初始位移C_C ……………………………………………………37
表4-9:樑結構受初始縱向均勻負載時之無因次化自然振動頻率初始應力+初始位移C_F ……………………………………………………39
表4-10:樑結構受初始縱向均勻負載時之無因次化自然振動頻率初始應力+初始位移C_P ……………………………………………………41
圖1-1：尤拉樑理論假設下之自由體圖 ……………………………4
圖1-2：Timoshenko樑理論假設下之自由體圖 ……………………7
圖1-3：高階樑理論假設下之自由體圖 ……………………………9
圖2-1：樑的尺寸標記和卡式系統之位移座標示意圖 …………13
圖2-2：變型前八節點原素圖 ………………………………………19
圖2-3：變型後八節點原素圖 ………………………………………19
圖3-1：位移邊界條件 ……………………………………………22
圖 4-1：P-P邊界條件下受軸向拉力和壓力示意圖 ………………26
圖 4-2：只考慮初始應力之無因次自然頻率曲線圖Umax=1…
…………………………………………………………………………43
圖 4-3：只考慮初始應力之無因次自然頻率曲線圖Umax=2…
…………………………………………………………………………43
圖 4-4：只考慮初始應力之無因次自然頻率曲線圖Umax=3…
…………………………………………………………………………43
圖 4-5：同時考慮初始應力與初始位移之無因次自然頻率曲線圖Umax=1……………………………………………………………44
圖 4-6：同時考慮初始應力與初始位移之無因次自然頻率曲線圖Umax=2…………………………………………………………44
圖 4-7：同時考慮初始應力與初始位移之無因次自然頻率曲線圖Umax=3……………………………………………………………44

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