本文係研究彈性支撐壓電複合層板之非線性振動分析。首先採用簡單高階剪變形理論，配合Von Karman大變形理論和壓電理論，利用漢姆頓定理（Hamilton’s Principle）和變分法推導出壓電複合層板之統御方程式，並利用雙重傅利葉級數（Double Fourier Series），推導出一組非線性代數方程組，然後利用Ritz Method求得非線性頻率解。
本文詳細探討不同振幅、不同長寬比、材料厚度、壓電材料疊層角度、邊界彈性係數等參數對非線性頻率的影響，所得部分結果與文獻比較十分吻合，可證實本文分析結果應屬正確，可提供壓電複合層板非線性振動分析參考之用。

The thesis studies the nonlinear vibrations of piezoelectric composite plates with elastically restrained boundary conditions. Based on the simple high-order theory, von Karman theory, and piezoelectric theory, the equations of motion for piezoelectric composite plates are derived by Hamilton’s principle and variational principle. Generalized double Fourier series are used to constitute a set of simultaneous algebraic nonlinear equations, and then the nonlinear frequencies are solved by Ritz method. 
    The influences of aspect ratio, thickness-to-length, angle of orientation, elastically restrained coefficients on nonlinear frequencies are particularly discussed in this thesis. A good agreement is obtained from the comparison of some special results between the thesis and literatures. It is proved that the results are correct in this thesis, and should serve as references for nonlinear vibration analysis of piezoelectric composite plates.

中文摘要 ------------------------------------------------------------------------- I
英文摘要 ------------------------------------------------------------------------ II
目    錄 ----------------------------------------------------------------------- III
圖表索引 ----------------------------------------------------------------------- V
第一章 緒論 -------------------------------------------------------------------- 1
1.1 前言 -------------------------------------------------------------------- 1
1.2 研究動機與目的 ----------------------------------------------------- 2 
1.3 文獻回顧 -------------------------------------------------------------- 3 
1.4 論文架構 -------------------------------------------------------------- 5 
第二章 基本理論 -------------------------------------------------------------- 7
2.1 壓電理論 -------------------------------------------------------------- 7
2.1.1 壓電效應 ------------------------------------------------------ 7
2.1.2 介電效應 ------------------------------------------------------ 7
2.1.3 壓電本構方程式 --------------------------------------------- 7
2.2 簡化高階壓電複合層板理論 -------------------------------------- 8
2.2.1 位移假設 ------------------------------------------------------ 8
2.2.2 應變與位移關係式 ----------------------------------------- 11
2.2.3 電場與位移關係式 ----------------------------------------- 12
2.2.4 基層座標與系統關係式 ----------------------------------- 12
第三章 能量法推導與頻率分析 ------------------------------------------- 17
3.1 統御方程式 --------------------------------------------------------- 17
3.2 邊界條件 ------------------------------------------------------------ 31
3.3 理論分析 ------------------------------------------------------------ 34
3.4 線性分析 ------------------------------------------------------------ 53
3.5 非線性分析 --------------------------------------------------------- 55
第四章 分析結果與討論 ---------------------------------------------------- 60
4.1 參考文獻之比較 --------------------------------------------------- 60
    4.1.1單層壓電板 -------------------------------------------------- 60
    4.1.2壓電複合層板 ----------------------------------------------- 62
4.2 彈性邊界之影響 --------------------------------------------------- 62
4.3 厚度之影響 --------------------------------------------------------- 63
4.4 不同長寬比之影響 ------------------------------------------------ 64 
4.5 複合角度之影響 --------------------------------------------------- 65
第五章 結論與展望 ---------------------------------------------------------- 78
5.1 結論 ------------------------------------------------------------------ 78
5.2 未來展望 ---------------------------------------------------------¬--- 79
參考文獻 ----------------------------------------------------------------------- 80
符號索引 ----------------------------------------------------------------------- 83
圖2-1 正壓電效應 ----------------------------------------------------------- 15
圖2-2 逆壓電效應 ----------------------------------------------------------- 15
圖2-3 介電效應 -------------------------------------------------------------- 16
圖2-4 壓電複合層板示意圖 ----------------------------------------------- 16
圖3-1 彈性邊界示意圖 ----------------------------------------------------- 59
圖4-1 彈性邊界對單層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/b=1;a/h=10） ------------------------------------------------------- 70
圖4-2 彈性邊界對兩層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------   （a/b=1;a/h=10） ------------------------------------------------------- 70
圖4-3 彈性邊界對三層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/b=1;a/h=10） ------------------------------------------------------- 71
圖4-4 彈性邊界對十層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/b=1;a/h=10） ------------------------------------------------------- 71
圖4-5 a/h對單層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/b=1;  ） --------------------------------------- 72
圖4-6 a/h對單層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/b=1;  ） ---------------------------------------- 72
圖4-7 a/h對單層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/b=1;  ） -------------------------------------- 73
圖4-8 a/h對單層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/b=1;  ） -----------------------¬---------------- 73
圖4-9 a/b對單層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/h=10;  ） ------------------------------------ 74
圖4-10 a/b對單層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/h=10;  ） ---------------------¬------------¬---- 74
圖4-11 a/b對單層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/h=10;  ） -----------------¬------------------ 75
圖4-12 a/b對單層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/h=10;  ） ---------------¬--------------------- 75
圖4-13 疊層角度對單層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/h=1; a/h=10; ） --------------¬---------------- 76
圖4-14 疊層角度對兩層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/h=1; a/h=10; ） ----------¬-------------------- 76
圖4-15 疊層角度對三層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/h=1; a/h=10; ） ----------¬-------------------- 77
圖4-16 疊層角度對十層壓電板非線性頻率的影響分析圖
-------------------------------------------------------------------------------------------------------------------------（a/h=1; a/h=10; ） ------------¬------------------ 77
表4-1 理論解與參考文獻[12]之對照 ------------------------------------ 67
表4-2 材料參數 -------------------------------------------------------------- 68
表4-3 理論解與參考文獻[6]之對照 -------------------------------------- 69
表4-7 理論解與參考文獻[17]之對照 ------------------------------------ 69

1.	Smart Materials : A Technical and Market Assessment, BCC Research (2006).
2.	周卓明，壓電力學，全華科技圖書股份有限公司，2003年。
3.	C. Y. Chia, Nonlinear Analysis of Plates, New York : McGraw-Hill, (1980).
4.	Z. Ding, “Natural Frequencies of elastically restrained rectangular plates using a set of static beam functions in the Rayleigh-Ritz method,” Computers and Structure, Vol. 57, pp.731-735 (1995).
5.	Z. Ding, Y. K. Cheung, F. T. K. Au and S. H. Lo, “Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method,” International Journal of Solids and Structures, Vol. 39, pp.6339-6353 (2002).
6.	P. Heyliger and D. A. Saravanos, “Exact free-vibration analysis of laminated plates with embedded piezoelectric layers,” Acoustical Society of America, Vol. 98, pp.1547-1557 (1995).
7.	A. Benjeddou and J. Francois, “A two-dimensional closed-form solution for the free-vibrations analysis of piezoelectric sandwich plates,” International Journal of Solids and Structures, Vol. 39, pp.1469-1486 (2002).
8.	X. H. Wu, C Chen, Y. P. Shen and X. G. Tian, “A high order theory for functionally graded piezoelectric shells” International Journal of Solids and Structures, Vol. 39, pp.5325-5344 (2002).
9.	U. V. Wagner, “Non-linear longitudinal vibrations of piezoceramics exicited by weak electric fields,” International Journal of Non-Linear Mechains, Vol. 38, pp.565-574 (2003).
10.	X. L. Hung and H. S. Shen, “Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators,” International Journal of Mechanical Sciences, Vol. 47, pp.187-208 (2005).
11.	S. K Parashar, U. V. Wagner and P. Hagdorn, “Nonlinear shear-induced flexural vibrations of piezoelectric actuators:experiments and modeling,” International Journal of Sound and Vibration, Vol. 285, pp.959-1014 (2005)
12.	P. Cupial, “Three-dimensional natural vibration analysis and energy considerations for a piezoelectric rectangular plate,” International Journal of Sound and Vibration, Vol. 283, pp.1093-1113 (2005).
13.	X. Shu, “Free vibration of laminated piezoelectric composite plates based on an accurate theory,” Composite structures, Vol. 67, pp.375-382 (2005).
14.	Z. Zhang, C. Feng and K. M. Liew, “Three-dimensional vibration analysis of mulitilayered piezoelectric composite plates,” International Journal of Engineering Science, Vol. 44, pp.397-408 (2006).
15.	P. Topdar, A. H. Sheikh and N. Dhang, “Vibration characteristics of composite/sandwich lamonates with piezoelectric layers using a refined hybrid plate model,” International Journal of Mechanical Science, Vol. 49, pp.1193-1203 (2007).
16.	S. N. Mahmoodi and N. Jalili, “Non-linear vibrations and frequency response analysys of piezoelectrically driven microcantilevers” International Journal of Non-Linear Mechains, Vol. 42, pp.577-587 (2007).
17.	K. Jayakumar, D. Yadav and B. N. Rao, “Nonlinear free vibration analysis of simply supported piezo-laminated plates with random actuation electric potential difference and material properties” nonlinear science and numerical simulation(2008).
18.	鍾曜光，〝壓電式微超音波感測器〞，碩士論文，國立台灣大學機械工程學研究所，2001年。
19.	林憲陽，〝壓電陶瓷複合層板動態特性之數值分析與實驗量測〞，博士論文，國立台灣大學機械工程學研究所，2002年。
20.	施建宇，〝壓電複合材料板的振動探討及其應用在超音波馬達之初步研究〞，碩士論文，國立成功大學土木學研究所，2003年。
21.	李東旻，〝具壓電加強環之雙跨距圓柱薄殼結構之振動分析〞，碩士論文，國立成功大學工程科學研究所，2006年。
22.	P. Topdar, A. H. Sheikh and N. Dhang, “Vibration characteristics of composite/sandwich laminates with piezoelecteic layers using a refind hybrid plate model,” International Journal of Mechanical Sciences, Vol. 49, pp.1193-1203 (2007). 
23.	George Lindfield John Penny, Numerical Methods Using Matlab, 全華科技股份有限公司，2003年。