臺灣博碩士論文加值系統

本文係研究承受電場或電壓致動且邊界為旋轉彈性支撐之壓電複合層板。首先採用簡化高階複合層板理論，配合壓電理論，並利用廣義雙重傅利葉級數法(The Generalized Double Fourier Series Method)將位移與電位函數假設成滿足邊界條件之傅利葉級數，對壓電複合層板之能量利用漢米爾頓定理分析，由此推導出一組非線性代數方程組逐步求解，以其了解壓電複合層板在承受電場或電壓致動時之行為。本文詳細探討各種不同之長寬比、材料係數、材料厚度、壓電材料疊層角度、邊界彈性係數、電場以及電壓等參數對非線性彎曲的影響，部分結果與文獻比較十分吻合。本研究成果將可提供壓電複合層板設計者和分析者參考之用。

A simple high-order theory is presented to examine the nonlinear analysis of the piezoelectric composite plates subjected electric field or voltage with elastically restrained boundary conditions in this thesis. The simple high-order theory employs parabolic shear strain variation across the thickness, and requires no correction factors because it satisfies shear free surface boundary conditions. Generalized double Fourier series are used to constitute a set of nonlinear algebra equations which solution is obtained by numerical method.The influences of aspect ratio, material properties, thickness-to-length, angle of orientation, elastically restrained coefficients, electric field, and voltage are particularly discussed in this thesis. Some special cases are compared with those given in the literature and good agreement is obtained.The research results should serve as references for designer and analysis of piezoelectric composite plates.

中文摘要-------------------------------------------------------------------------------------------------------------I
英文摘要------------------------------------------------------------------------------------------------------------II
目 錄-----------------------------------------------------------------------------------------------------------------III
表 目 錄--------------------------------------------------------------------------------------------------------------V
圖 目 錄-------------------------------------------------------------------------------------------------------------VI
符號說明----------------------------------------------------------------------------------------------------------VIII
第一章 緒論----------------------------------------------------------------------------------------------------------1
1.1前言----------------------------------------------------------------------------------------------------------------1
1.2文獻回顧----------------------------------------------------------------------------------------------------------2
1.3研究動機----------------------------------------------------------------------------------------------------------5
1.4研究方法----------------------------------------------------------------------------------------------------------5
第二章 壓電理論----------------------------------------------------------------------------------------------------8
2.1壓電效應----------------------------------------------------------------------------------------------------------8
2.2介電效應----------------------------------------------------------------------------------------------------------8
2.3壓電本構方程式-------------------------------------------------------------------------------------------------8
第三章 簡化高階壓電複合層板理論-------------------------------------------------------------------------11
3.1位移假設---------------------------------------------------------------------------------------------------------11
3.2應變與位移關係式--------------------------------------------------------------------------------------------14
3.3電場與電位關係式--------------------------------------------------------------------------------------------15
3.4積層座標系統關係式-----------------------------------------------------------------------------------------15
第四章 能量法推導與理論分析-------------------------------------------------------------------------------19
4.1統御方程式------------------------------------------------------------------------------------------------------19
4.2電位與電場轉換之統御方程式-----------------------------------------------------------------------------34
4.3邊界條件----------------------------------------------------------------------------------------------------------44
4.3.1支撐邊界--------------------------------------------------------------------------------------------------------44
4.3.2電性邊界--------------------------------------------------------------------------------------------------------46
4.4 理論分析----------------------------------------------------------------------------------------------------------47
4.4.1靜態分析--------------------------------------------------------------------------------------------------------47
4.4.2由電位致動之靜態分析--------------------------------------------------------------------------------------63
4.4.3由電場致動之靜態分析--------------------------------------------------------------------------------------64
第五章 分析結果與討論--------------------------------------------------------------------------------------------69
5.1 固定電場致動分析----------------------------------------------------------------------------------------------69
5.1.1參考文獻之比較與收斂性-----------------------------------------------------------------------------------69
5.1.2彈性邊界之影響------------------------------------------------------------------------------------------------74
5.1.3壓電材料複合角度之影響-----------------------------------------------------------------------------------83
5.2 非固定電場致動分析-------------------------------------------------------------------------------------------86
5.2.1非固定電場致動之線性與非線性比較-------------------------------------------------------------------86
5.3 電壓致動分析----------------------------------------------------------------------------------------------------90
5.3.1電壓增加與壓電材料之影響--------------------------------------------------------------------------------91
5.3.2厚度與壓電材料之影響--------------------------------------------------------------------------------------93
5.3.3長寬比與厚度之影響------------------------------------------------------------------------------------------95
5.4 固定電壓下受均佈負荷之影響-------------------------------------------------------------------------------96
5.5 電場假設與線性電壓假設以及曲線電壓假設致動比較-----------------------------------------------99
第六章 結論與建議--------------------------------------------------------------------------------------------------102
6.1 結論-----------------------------------------------------------------------------------------------------------------102
6.2 建議-----------------------------------------------------------------------------------------------------------------103
參考文獻----------------------------------------------------------------------------------------------------------------105

表目錄
表5-1 正方板理論解與參考文獻之對照--------------------------------------------------------------------------71
表5-2 長方板理論解與參考文獻之對照--------------------------------------------------------------------------72
表5-3 彈性邊界線性理論解------------------------------------------------------------------------------------------74
表5-4 彈性邊界非線性理論解---------------------------------------------------------------------------------------75
表5-5 不同壓電疊交角度之非線性解比較表--------------------------------------------------------------------83
表5-6 不同疊交角度與彈性邊界比較表--------------------------------------------------------------------------85
表5-7 電壓與壓電材料之影響比較表-----------------------------------------------------------------------------92
表5-8 厚度與壓電材料之影響比較表-----------------------------------------------------------------------------93
表5-9 固定電壓下長寬比與厚度之比較表(b=0.3m) ----------------------------------------------------------95
表5-10 線性電壓與固定電場分析結果比較表-----------------------------------------------------------------100
表5-11 線性電壓與拋物線電壓分析結果比較表( =200V) -------------------------------------------------101

圖目錄
圖1-1 古典板理論示意圖-----------------------------------------------------------------------------------------------2
圖1-2 一階剪變形理論示意圖-----------------------------------------------------------------------------------------2
圖1-3 高階剪變形理論示意圖-----------------------------------------------------------------------------------------3
圖2-1 壓電效應示意圖---------------------------------------------------------------------------------------------------8
圖2-2 介電效應示意圖---------------------------------------------------------------------------------------------------8
圖5-1 參考文獻[6]之結構示意圖-----------------------------------------------------------------------------------71
圖5-2 雙重傅利葉級數項次之收斂圖-----------------------------------------------------------------------------73
圖5-3 雙重傅利葉級數項次之收斂圖-----------------------------------------------------------------------------73
圖5-4 = = = =0之厚度-位移關係圖(a) ---------------------------------------------------------------------------76
圖5-4 = = = =0之厚度-位移關係圖(b) ---------------------------------------------------------------------------76
圖5-5 = = = =0.1之厚度-位移關係圖(a) -------------------------------------------------------------------------77
圖5-5 = = = =0.1之厚度-位移關係圖(b) -------------------------------------------------------------------------77
圖5-6 = = = =1之厚度-位移關係圖(a) ---------------------------------------------------------------------------78
圖5-6 = = = =1之厚度-位移關係圖(b) ---------------------------------------------------------------------------78
圖5-7 = = = = 之厚度-位移關係圖(a) ----------------------------------------------------------------------------79
圖5-7 = = = = 之厚度-位移關係圖(b) ----------------------------------------------------------------------------79
圖5-8 = = = = 1、0.1、0.2、0.3之厚度-位移(線性與非線性)重疊比較圖----------------------------80
圖5-9 = =0.1、1、10、 之 = -位移關係圖-----------------------------------------------------------------------82
圖5-10 壓電材料疊交角度之位移曲線關係圖------------------------------------------------------------------84
圖5-11 z方向電場-位移(線性與非線性)比較圖(a) ------------------------------------------------------------87
圖5-11 z方向電場-位移(線性與非線性)比較圖(b) ------------------------------------------------------------87
圖5-11 z方向電場-位移(線性與非線性)比較圖(c) ------------------------------------------------------------88
圖5-11 z方向電場-位移(線性與非線性)比較圖(d) ------------------------------------------------------------89
圖5-12 結構示意圖------------------------------------------------------------------------------------------------------90
圖5-13 電壓與壓電材料之關係圖-----------------------------------------------------------------------------------92
圖5-14 厚度與壓電材料之關係圖-----------------------------------------------------------------------------------94
圖5-15 固定電壓下長寬比與厚度之關係圖(b=0.3m) --------------------------------------------------------95
圖5-16 固定電壓下受均佈負荷之關係圖(b/h=200) -----------------------------------------------------------96
圖5-17 固定電壓下受均佈負荷之關係圖(b/h=300) -----------------------------------------------------------97
圖5-18 固定電壓下受均佈負荷之關係圖(b/h=400) -----------------------------------------------------------97
圖5-19 不同厚度於固定電壓下受均佈負荷之比較圖---------------------------------------------------------98
圖5-20 不同材料於固定電壓下受均佈負荷之比較圖---------------------------------------------------------98

1. C. Y. Chia, Nonlinear Analysis of Plates, New York : McGraw-Hill, (1980)
2. J. N. Reddy, Energy and Variational Methods in Applied Mechanics : With An Introduction to The Finite Element Method, New York : Wiley, (1984)
3. I. H. SHAMES and C. L. DYM, Energy and Finite Element Methods in Structural Mechanics, Hemisphere Publishing,(1985)
4. 張宏政，〝邊界限制有缺陷複合層板承受衝擊負荷之動態分析〞，碩士論文，私立淡江大學機電 工程學系研究所，1996年
5. 周卓明，壓電力學，全華科技圖書股份有限公司，2003年
6. K. D. Jonnalagadda, G. E. Blandford and T. R. Tauchert,“Piezothermoelastic Composite Plate Analysis Using First-Order Shear Deformation Theory, ”Computers & structures, Volume:51, 79-89, (1994)
7. H. S. Shen, “Nonlinear Bending Analysis of Unsymmetric Cross-Ply Laminated Plates With Piezoelectric Actuators in Thermal Environments,” Composite Structures, Volume:63, 167- 177,(2004)
8. J. H. Huang and T. L. Wu,“Analysis of Hybrid Multilayered Piezoelectric Plates,”International Journal of Engineering Science,Volume: 34, 171-181, (1996)
9. 游�琱@ ，〝壓電薄板材料之靜態與動態分析〞，碩士論文，私立逢甲大學機械工程學系研究所 ，1999年
10. A. Fernandes and J. Pouget, “Two-Dimensional of Laminated Piezoelectric Composites:Analysis and Numerical Results,” Thin-Walled Structures, Volume: 39, 3-22, (2001)
11. A. Fernandes and J. Pouget, “An Accurate Modelling of Piezoelectric Multi-Layer Plates,” European Journal of Mechanics A/Solids hin-Walled Structures, Volume: 21,629-651, (2002)
12. X. H. Wu, C. Chen, Y. P. Shen, X. G. Tian, “A high order theory for functionally graded piezoelectric shells,” International Journal of Solids and Structures Volume: 39, 5325-5344, (2002)
13. S. Shen, Z. B. Kuang, “An Active Control Model of Laminated Piezothmoelastic Plate, ” International Journal of Solids and Structures Volume: 36,1925-1947,(1999)
14. 李冠學，〝壓電平板之模擬與量測〞，碩士論文，國立臺灣大學應用力學研究所 ， 2001年
15. 黃吉宏，〝應用振幅變動電子斑點干涉術探討三維壓電材料體及含裂紋板的振動問 題〞，博士論文，國立台灣大學機械工程學系研究所，1998年
16. 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 臺灣大學機械工程學研究所，2002年
17. 林忠義，〝壓電材料應用於複合樑的之動態穩定性分析〞，碩士論文，國立成功大 學機械工程學系研究所，2000年
18. J. H. Han and I. Lee, “Analysis of Composite Plates with Piezoelectric Actuators for Vibration Control Using Layerwise Displacement Theory,” Composites Part B: Engineering ,Volume: 29, 621-632, (1998)
19. L. W. Chen, C. Y. Lin and C. C. Wang, “Dynamic Stability Analysis and Control of A Composite Beam with Piezoelectric Layers,” Composite Structures ,Volume: 56, 97-109, (2002)
20. C. H. Nguyen, S. J. Pietrzko, “Piezoelectric-Mechanical-Acoustic Couplings from A PZT- Actuated Vibrating Beam and Its Sound Radiation,” Mechanical Systems and Signal Processing ,Volume: 18, 929-945,(2004)
21. J. H. Huang and H. I. Yu, “Dynamic Electromechanical Response of Plates as Sensors or Actuators,” Materials Letters, Volume: 46, 70-80, (2000) )