本研究以理論解析雙片PZT壓電薄板與封閉可壓縮流體耦合問題，並應用能量法與力學平衡法分析圓柱與長方柱流-固耦合性系統的振動特性。第一部分討論的圓柱流-固耦合系統，流體為非黏滯、可壓縮的圓柱封閉流場，而固體的部分則為兩片壓電圓形薄板，以相同邊界接觸圓柱流場雙側，分別以全固定(clamped-edge)及全自由(free-edge)兩種邊界條件構成兩種流-固耦合系統，首先固體的位移函數計算應用壓電材料力學理論與力平衡方程式推導壓電薄板的統御方程式，流體的計算則是應用聲場方程式搭配流-固邊界連續條件求得流體受固體影響的運動行為，最後使用能量法計算圓柱流-固耦合系統的共振頻率(resonant frequency)、振動模態(mode shape)與流場壓力(pressure field)。第二部分討論的長方柱流-固耦合系統，流體亦為非黏滯、可壓縮的封閉流場，而固體部分則改為兩片壓電懸臂板，其理論模型將兩片壓電懸臂板以單邊固定之邊界條件置於長方柱封閉流場內的對稱位置，固體計算採用壓電薄板統御方程式與疊加板原理，流體的計算採用聲場方程式搭配流-固邊界連續條件求得流體受固體影響的運動行為，最後使用力學平衡法計算長方柱流-固耦合系統的共振頻率、振動模態與流場壓力。所有理論解析的結果使用有限元素軟體確立準確性，本研究所獲得結果亦探討流體特性與深度對流-固耦合振動行為的影響。

This study investigated the vibration characteristics of two piezoelectric plates coupled with bounded compressible inviscid fluid by theoretical analysis. The Rayleigh–Ritz method and the coupled equation of hydrostatic equilibrium were developed to study two kinds of Fluid-Structure Interaction (FSI) system. The first FSI system in this study is two piezoelectric circular plates placed on the top and the bottom of a cylindrical container, respectively. The governing equation of piezoelectric circular plate is obtained by using the constitutive equations of piezoelectric plate and the equilibrium of mechanic. The governing equation of fluid is obtained from acoustic wave equation. The FSI between the piezoelectric plate and bounded fluid are obtained by employing the integral transformation technique. The frequency eigenfuction of the FSI system can be derived by the Rayleigh–Ritz method. Finally, the dynamic characteristic of the FSI system, including resonant frequencies, corresponding mode shapes, and pressure field of the fluid can be obtained from theoretical solution. The second FSI system in this study is two piezoelectric retangular cantilever plates immersed in the wall of rectangular container which is symmetric to the middle plane of container. The vibration displacement of piezoelectric retangular cantilever plate is obtained by the governing equation of piezoelectric plate and the superposition method. The governing equation of fluid is obtained from acoustic wave equation. The FSI between the piezoelectric plate and fluid are obtained by employing the integral transformation technique. The frequency eigenfuction of the FSI system can be derived by the coupled equation of hydrostatic equilibrium. Resonant frequencies, corresponding mode shapes, and pressure field of the fluid can be obtained by solving the characteristics equation.The vibration characteristics of theoretical analysis are verified with the results from finite element method(FEM). Furthermore, the theoretical solution was used to discuss the effects of the compressibility, density, depth of fluid on the resonant frequency.

[1]S. S. Rao, “Vibration of Continuous Systems”, Wesley, Coral Gables, Florida, 2007.
[2]A. W. Leissa, “The free vibration of rectangular plates”, Journal of Sound and Vibration, vol. 31, pp. 257-293, 1973
[3]D. J. Gorman, “Free vibration analysis of cantilever plates by the method of superposition”, Journal of Sound and Vibration, vol. 49, pp.453-467, 1976
[4]D. Avalos, P. A. A. Laura, R. H. Gutierrez, “On the influence of the poisson ratio on the nature frequency of free circular plates”, Journal of Sound and Vibration, vol. 132, pp.341-345, 1989
[5]H. F. Tiersten, “Linear Piezoelectric Plate Vibrations”. New York: Plenum, 1969.
[6]R. D. Mindlin, “High frequency vibrations of piezoelectric crystal plates”, International Journal of Solids and Structures, vol. 8, 1972
[7]P. R Heyliger, G. Ramirez, “Free vibration of laminated circular piezoelectric plates and discs”, Journal of Sound and Vibration, vol.229, pp.935-956, 2000
[8]吳亦莊，理論解析與實驗量測壓電平板的面外振動及特性探討，中華民國98年
[9]M. K. Kwak, K. C. Kim, “Axisymmetric vibration of circular plates in contact with fluid”, Journal of Sound and Vibration, vol.146, pp.381-389, 1990
[10]M. Amabili, A. Pasqualini, and G. Dalpiaz, “Natural frequencies and modes of free-edge circular plates vibrating in vacuum or in contact with liquid”, Journal of Sound and Vibration, vol. 188, pp. 685-699, 1995.
[11]M. Amabili and M. K. Kwak, “Free vibrations of circular plates coupled with liquids: Revising the lamb problem”, Journal of Fluids and Structures, vol. 10, pp. 743-761, 1996.
[12]M. K. Kwak, “Hydroelastic vibration of circular plates”, Journal of Sound and Vibration, vol. 201, pp. 293-303, 1997.
[13]C. C. Liang, C. C. Liao, Y.S. Tai, W. H. Lai, “The free vibration analysis of submerged cantilever plates.” Ocean Engineering, vol. 28, pp. 1225–1245, 2000
[14]M. Amabili, "Effect of finite fluid depth on the hydroelastic vibrations of circular and annular plates." Journal of Sound and Vibration, vol. 193, pp. 909-925, 1995
[15]K.H. Jeong, “Free vibration of two identical circular plates coupled with bounded fluid”, Journal of Sound and Vibration, vol. 260, pp. 653-670, 2003
[16]K.H. Jeong and K.J Kim, “Hydroelastic vibration of a circular plate submerged in a bounded compressible fluid”, Journal of Sound and Vibration, vol. 283, pp. 153-172, 2005
[17]C.Y. Liao and C.C. Ma, “Vibration characteristics of rectangular plate in compressible inviscid fluid”, Journal of Sound and Vibration, vol. 362, pp. 228-251, 2016
[18]閻建佑，應用於幫浦結構之對稱壓電雙層圓盤以同反相驅動的固液耦合振動特性分析與流率量測，中華民國104年
[19]廖展誼，壓電圓板耦合可壓縮流體之動態特性研究，中華民國102年
[20]廖展誼，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，中華民國105年
[21]L.E Kinsler, A.R. Frey, A.B. Coppens, J.V. Sanders, “Fundamentals of Acoustics 4th Edition”, John Wiley & Sons,Inc, 2000