基於平板預加應力振動研究一般以均勻分佈力為主，因此本文將探討施以正弦函數分佈力的平板振動問題。研究方法是以一階剪切變形板理論為基礎，使用有限元素法分析其振動頻率之變化。結果得知對於邊界條件為CCCC、SSSS、CFCF、SFSF之平板，施以對稱正弦函數分佈之預加負載於兩相對邊的時候，頻率會隨著模態數增加而上升而後趨緩。增加正弦波數目時，頻率會先短暫上升後下降，此種頻率的變化只有當邊界條件為SFSF的低模態振動時，才比較明顯。

Due to most studies on vibration of pre-loaded rectangular plate being subject to uniform loading, this thesis will investigate vibration of plate under preloading with sine functional distributions.
The approach behind this study is using first-order shear deformation plate theory and finite element method to analyze vibration frequency.
Study results show that, when a plate’s boundary condition is CCCC, SSSS, CFCF, or SFSF with sine functional pre-loading, its vibration frequency will increase to an extent with little difference when the number of modes increases. Vibration frequency will increase shortly then decrease when increasing the number of sine waves. Vibration frequency will also increase when increasing stress parameters. However, obvious frequency changes are observed only at lower modes with SFSF boundary condition.

第一章 緒論 ．．．．．．．．．．．．．．．．．．．．．．．1

1.1前言．．．．．．．．．．．．．．．．．．．．．．．1

1.2文獻回顧．．．．．．．．．．．．．．．．．．．．．2

1-2-1古典板理論．．．．．．．．．．．．．．．．．2

1-2-2 一階剪切變形板理論．．．．．．．．．．．．．4

1-2-3高階平板理論．．．．．．．．．．．．．．．．5

1-2-4三維彈性力學．．．．．．．．．．．．．．6

第二章 矩形版之振動分析理論．．．．．．．．．．．．7

2.1前言．．．．．．．．．．．．．．．．．．．．．．7

2.2自由振動有限元素列式．．．．．．．．．．．．7

2.3受負載之振動有限元素列式．．．．．．．．．12

2.4挫曲．．．．．．．．．．．．．．．．．．．．15

2.5均勻與不均勻施力分佈．．．．．．．．．．．．．17


第三章 問題解析．．．．．．．．．．．．．．．．．．．21

3.1前言．．．．．．．．．．．．．．．．．．．．21

3.2問題描述．．．．．．．．．．．．．．．．．．21

第四章 結果與討論．．．．．．．．．．．．．．．．．．26

4.1前言．．．．．．．．．．．．．．．．．．．．26

4.2文獻比較．．．．．．．．．．．．．．．．．．26

4.3非均勻預加負載．．．．．．．．．．．．．．．．28

第五章 結論與建議．．．．．．．．．．．．．．．．．．65

5.1結論．．．．．．．．．．．．．．．．．．．．65

5.2建議．．．．．．．．．．．．．．．．．．．．65

參考文獻．．．．．．．．．．．．．．．．．．．．．．．．．67

附 錄．．．．．．．．．．．．．．．．．．．．．．．．．72

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