本論文結合理論解析、有限元素模擬與實驗量測，探討陶瓷壓電雙晶片於不同邊界下的面外振動特性。理論解析首先利用線性壓電理論與力與力矩平衡條件式，將壓電三維力電耦合模型等效為通用矩形平板參數，接著分別利用一維樑理論與二維樑函數模型推導壓電樑於各邊界下面外振動特性，其探討邊界有全固定邊界、單邊固定、全自由、彈簧邊界，其中彈簧邊界針對線性彈簧與扭轉彈簧進行理論解析，其中以樑函數建構二維長方薄板之振動模態，最後以瑞雷-瑞茲求得壓電薄板分析共振之頻率和位移模態，並針對撓性邊界探討壓電薄板於不同線性勁度與扭轉勁度下之共振頻率變化走向，所有理論解析結果皆與有限元素分析數值計算結果進行比較，驗證理論分析之準確性與適用範圍。實驗量測方面使用電子斑點干涉術(AF-ESPI)針對壓電材料進行即時動態量測紀錄模態振形與共振頻率，結合穩態掃頻之雷射都卜勒振動儀(LDV)量測結果，理論分析皆符合實驗量測與有限元素數值計算結果，且由理論解析與實驗結果得知撓性邊界相較於固定邊界激發更低共振頻率的剛性參數設計條件，未來可應用可變剛性結構於能量擷取系統獲取最佳效率。

This study thoroughly analyzed to out-of-plane vibration of piezoelectric plate in several boundary conditions, based on theoretical analysis, finite element method (FEM) and experimental measurements. The Euler-Bernoulli beam theory is used to derive the vibration characteristics of the piezoelectric beam under different boundary conditions. The vibration mode of two-dimensional rectangular plate is analyzed by the beam function. Then the Rayleigh-Ritz method is used to determine the resonant frequency and vibration mode of the piezoelectric plate. The fully-clamped (CC), clamped-free (CF), fully-free (FF), spring-free (S+SrF) boundary conditions are determined theoretically, and the flexible support is discussed on the spring boundary both in the linear and the torsional springs, respectively. The finite element method is used to mainly verify on the correspondent resonant frequency of the thin plate under different linear stiffness and torsion stiffness. The experimental measurements are used to obtain the dynamic characteristics of piezoelectric materials. The full-filed optically technique, called as amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), and the Laser Doppler Vibrometer (LDV) are used to obtain the vibration properties of piezoelectric specimen under CF and S+SrF boundary conditions.
The theoretical analysis has good agreement with the experimental measurement and the finite element calculation. It showed that the flexible boundary condition used on piezoelectric plate has lower natural frequencies than the clamped-free boundary condition. The optimal design can be applied to variable stiffness piezoelectric energy harvesting system in approaching to the best efficiency in future.

[1] W. Ritz, “Theorie der Transversalschwingungen einer quadratischen Platte mit freien Randern.” Annalen der Physik 333(4),pp. 737-786,1909.
[2] D. Young, “Vibration of rectangular plates by the Ritz method.” Journal of Applied Mechanics-Transactions of the ASME 17(4),pp. 448-453,1950.
[3] G. B. Warburtion, “The vibration of rectangular plates.” Proceedings of the Institution of Mechanical Engineers 168(1),pp. 371-384,1954.
[4] A. W. Leissa, “Vibration of plates, NASA SP-160,(1969).” US Washington. W. Leissa. J Sound and Vibration 56,pp. 313,1978.
[5] G. B. Warburton, and S. L. Edney, “Vibrations of rectangular plates with elastically restrained edges.” Journal of Sound and Vibration, 95(4),pp. 537-552,1984.
[6] C. S. Kim, and. P. G. Young, and S. M. Dickinson, “On the flexural vibration of rectangular plates approached by using simple polynomials in the rayleigh-ritz method.” Journal of Sound and Vibration 143(3),pp. 379-394,1990.
[7] X. Zhang, and W. Li, “Vibrations of rectangular plates with arbitrary non-uniform elastic edge restraints.” Journal of Sound and Vibration, 326(1-2),pp. 221-234,2009.
[8] C. C. Liang, C. C. Liao, Y. S. Tai, and W. H. Lai, “The free vibration analysis of submerged cantilever plates.” Ocean Eingineering 28(9),pp. 1225-1245,2001.
[9] K. Khorshid and F. Sirwan, “Free vibration analysis of a laminated composite rectangular plate in contact with a bounded fluid.” Composite Structures 104,pp. 176-186,2013.
[10] 徐雪維, 馬劍清, “流-固耦合問題之流場中長方形薄板振動特性理論分析與數值探討”, 國立台灣大學機械工程研究所碩士論文,2015.

[11] 廖展誼, 馬劍清, “矩形平板於流固耦合問題的振動特性與暫態波傳之理論分析、數值計算與實驗量測”, 國立台灣大學機械工程研究所博士論文,2018.
[12] W. G. Cady, Piezoelectricity. McGraw-Hill Book Co. Inc., New York, 1946.
[13] W. P. Mason, Piezoelectric crystals and their application to ultrasonics. New York: Can Nostrand, 1950.
[14] H. F. Tiersten, Linear Piezoelectric Plate Vibrations. New York: Plenum, 1969.
[15] IEEE standard on piezoelectricity. IEEE Ultrasonics Ferroelectrics and Frequency Control Society, ANSI/IEEE Std 176-1987.
[16] H. S. Tzou, Piezoelectric shells: distributed sensing and control of continua. Kluwer Academic Publishers, 1993.
[17] 吳亦莊, 馬劍清, “理論解析與實驗量測壓電平板的面外振動及特性探討”, 國立台灣大學機械工程研究所碩士論文,2009.
[18] Y. H. Huang, and C. C. Ma, “Experimental measurements and finite element analysis of the coupled vibrational characteristics of piezoelectric shells,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 59(4), pp. 785-798, 2012.
[19] 林蕙君, 舒貽忠, ”串聯陣列式壓電振動子能量擷取系統之分析研究”,國立台灣大學應用力學研究所碩士論文,2012。
[20] Y. H. Huang, and C. C. Ma, Z. Z. Li, “Investigations on vibration characteristics of two-layered piezoceramic disks.” Journal of Solids and Structures 51,pp. 227-251,2014
[21] 周聖倫, 黃育熙, “壓電能量截取系統以邊界設計方法降低共振頻率之研究”, 國立台灣科技大學機械工程學系碩士論文,2017
[22] S. Y. Wang, “A finite element model for the static and dynamic analysis of a piezoelectric bimorph.” International Journal of Solids and Structures, 41, pp. 4075-4096, 2004
[23] C. C. Ma, Lin Y.C., Y. H. Huang, and H. Y. Lin, “Experimental measurement and numerical analysis on resonant characteristics of cantilever plates for piezoceramic bimorphs,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 54(2), pp. 227-239,2007.
[24] 黃育熙, 馬劍清, ” 壓電陶瓷平板、薄殼、與雙晶片三維耦合動態特性之實驗量測、數值計算、與理論解析”, 國立台灣大學機械工程研究所博士論文,2009
[25] 鄭雅倫, 趙振綱, 黃育熙, ”串聯型三層壓電雙晶片於電極設計搭配模態振形之能量截取效率提升”, 國立台灣科技大學機械工程學系碩士論文,2016
[26] J. N. Butters, and J. A. Leendertz,“Speckle pattern and holographic techniques in engineering metrology,” Optics and Laser Technology, 3(1), pp. 26-30, 1971
[27] K.Hgmoen, and O. J. Lkberg, ”Detection and measurement of small vibrations using electronic speckle pattern interferometry,”Applied Optics, 16(7), pp. 1869-1875, 1977
[28] C. Wykes, “Use of electronic speckle pattern interferometry (ESPI) in the measurement of static and dynamic surface displacements,” Optical Engineering, 21, pp. 400-406, 1982
[29] S. Nakadate, H. Saito, and T. Nakajima, “Vibration measurement using phase-shifting stroboscopic holographic interferometry,” Journal of Modern Optics, 33(10) , pp. 1295-1309, 1986
[30] W. C. Wang., C. H. Hwang and S. Y. Lin, “Vibration measurement by the time-averaged electronic speckle pattern interferometry methods,” Applied Optics, 35(22) , pp. 4502-4509, 1996
[31] C. C. Ma, and C. H. Huang, “Experimental and numerical analysis of vibrating cracked plates at resonant frequencies,” Experimental Mechanics, 41(1), 8-18,2001
[32] C. C.Ma, and C. H. Huang, “Experimental full field investigations of resonant vibrations for piezoceramic plates by an optical interferometry method,” Experimental Mechanics, 42(2) , pp. 140-146, 2002
[33] C. C. Ma, and Y. C. Lin,”Resonant vibration of piezoceramic plates in fluid.” Interaction and Multiscale Mechanics,1(2),pp. 177-190,2008
[34] C. C. Ma, H. Y. Lin, Y. C. Lin, and Y. H. Huang, “Experimental and numerical investigations on resonant characteristics of a single-layer piezoceramic plate and a cross-ply piezolaminates composite plate,” Journal of the Acoustical Society of America, 119(3) , pp. 1476-1486, 2006
[35] 林憲陽，馬劍清”應用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”,國立台灣大學機械工程研究所博士論文,1998年6月
[36] R. C. Hibbeler, Mechanics of materials. 3rd edition, Prentice Hall Press, 1997
[37] Proldv LDV-OFV505 Retrieved July 6,2019 form
(https://www.polytec.com/int/vibrometry-polytec/vibrometry-products-polytec/single-point-vibrometers/ofv-5000-modular-vibrometer/ )
[38] Proldv Laser Doppler optical system Retrieved July 6,2019 form
(https://www.polytec.com/int/vibrometry-polytec/vibrometry-products-polytec/single-point-vibrometers/ofv-5000-modular-vibrometer/ )
[39] Abaqus 6.14 coupling constranints. Retrieved July 6,2019 form
(https://www.sharcnet.ca/Software/Abaqus/6.14.2/v6.14/books/usb/default.htm?startat=pt08ch35s03aus133.html)