摘 要
本文主要利用有限元素法分析壓電三明治Mindlin板結構的自由振動特性；此結構分為上層下層為壓電材料，中間層一種的兩種不同的材料所組成。
本文運用Mindlin板理論定義位移、應變與應力，計算出單層板的應變能及動能，再以漢米爾頓原理求得單層Mindlin板的靜態平衡方程式，使用靜態的運動方程式推導出其位移場之解並表示成形狀函數(shape function)的形式。由應變能及位能找出質量矩陣及剛性矩陣，進而利用Lagrange’s equation將應變能及動能代入得到單層方板元素的運動方程式，然後依不同的邊界條件將元素結合，解出系統的模態頻率。進一步定義壓電三明治板的位移、應變，求出其質量矩陣及剛性矩陣和靜態運動方程式還有三明治Mindlin板的位移場通解，一樣利用Lagrange’s equation求得其模態頻率。最後加入適當電壓求其壓電三明治板整體位移。最後利用回授控制設計於抑制此結構的振動。

ABSTRACT

This study presents natural frequencies and vibration control of rectangular plate with single-layered、sandwich plate with piezoelectric by Finite Element Method.
Based on Mindlin’s plate theory, the displacements, strains and stresses of single-layered and sandwich plate can be defined to calculate strain energy and kinetic energy. The governing equation are formulated via the Hamilton’s principle. The displacements solved from static equilibrium equations are used as shape functions of one element. The Finite Element Method is employed to obtain the natural frequencies of the entire plate , and compare with analytic solution.
Then , we provide a constant voltage to show the displacement via its
static equilibrium equations. At least ,we use the vibration control system to design for investigating the displacement of the structure.

參考文獻

1.R. D. Mindlin, 1951, " Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates," Journal of Application Mechanics, Vol.18, pp. 31-38.
2.E. Reissner and Y. Stavsky, 1961, " Bending and Stretching of Certain Types of Heterogeneous Aeolotropic Elastic Plates”, "Journal of Application Mechanics”, Vol.28, pp. 402-408.
3.E. Reissner, 1945, " The Effect of Transverse Shear Deformation on the Bending of Elastic Plates, "Journal of Applied Mechanics, Vol. 12, pp.A69-77
4.P. C. Yang, C. H. Norris and Y. Stavsky, 1966 , “Elastic Wave Propagation in Heterogeneous Plates,” International Journal of Solids and Structures, Vol. 2, pp. 665-684.
5.J. M. Whitney and N. J. Pagano, 1970, “Shear Deformation in Heterogeneous Anisotropic Plates,” Journal of Applied Mechanics, Vol. 37, pp. 1031-1036.
6.R. D. Mindlin, A. Schacknow and H. Deresiewicz, 1956, “Flexural Vibration of Rectangular Plates,” Journal of Applied Mechanics, Vol. 23, pp. 431-436.
7.J. N. Reddy and N. D. Phan, 1985, “Stability and Vibration of Isotropic, Orthotropic and Laminated Plates According to a Higher-order Shear Deformation Theory,” Journal of Sound and Vibration, Vol. 98, pp. 157-170.
8.N. D. Phan and J. N. Reddy, 1985, “Analysis of Laminated Composite Plates Using a Higher-Order Shear Deformation Theory,” Journal of Numerical Methods in Engineering, Vol. 21, pp. 2201-2219.
9.A. A. Khdeir, 1988, “Free Vibration and Buckling of Symmetric Cross-Ply Laminated Plates by an Exact Method,” Journal of Sound and Vibration, Vol. 126, No. 3, pp. 447-461.
10.Aksu,G.(1984), ree vibration analysis of rectangular plates with cutout allowing for transverse shear deformation and rotary. Earthquake Engineering for Structural Dynamics, 12(5),pp709-714.
11.N. S. Putcha and J. N. Reddy, 1986, “Stability and Natural Vibration Analysis of Laminated Plates by Using a Mixed Element Based on Refined Plate Theory,” Journal of Sound and Vibration, Vol. 104, pp. 285-300.
12.A. K. Noor, 1973, “Free Vibrations of Multilayered Composite Plates,” AIAA Journal, Vol. 11, pp. 1038-1039.
13.J. N. Reddy, 1997, Mechanics of Laminated Composite Plates .Theory and Analysis, CRC Press, Boca Raton, FL.
14.L. X. Luccioni and S. B. Dong, 1998, “Levy-type Finite Element Analysis of Vibration and Stability of Thin and Thick Laminated Composite Rectangular Plates,” Composite Part B , Vol. 29, pp. 459-475.
15.C. C. Chao and Y. C. Chern, 2000, “Comparison of Natural Frequencies of Laminated by 3-D Theory,” Journal of Sound and Vibration, Vol. 230, No. 5, pp. 985-1007.
16.M. Aydogdu and T. Timarci, 2003, “Vibration Analysis of Cross-Ply Laminated Square Plates with General Boundary Condition,” Composites Science and Technology, Vol. 63, pp. 1061-1070.
17.K. M. Liew, K. C. Hung and M. K. Lim, 1993, “Method of Domain Decomposition in Vibration of Mixed Edge Anisotropic Plates,” International Journal of Solids and Structures, Vol. 30, No. 23, pp. 3281-3301.
18.T. P. Chang and M. H. Wu, 1997, “On the Use of Characteristics Orthogonal Polynomials in the Free Vibration Analysis of Rectangular Anisotropic Plates with Mixed Boundary and Concentrated Masses,” Computers and Structure, Vol.62, No. 4, pp. 699-713.
19.D. J. Dawe and D. Tan, 1999, “Finite Strip Buckling and Free Vibration Analysis of Stepped Rectangular Composite Laminated Plates,” International Journal for Numerical Methods in Engineering, Vol. 46, pp. 1313-1334.
20.G. Bradfiled“Ultrasonic transducers 1. Introduction transducers. Part A.”Ultrasonic,pp.112-113,Apr 1970.
21.Jin H.Hung ,and Heng-I Yu, “Dynamic electromechanical response of piezoelectric plates as sensors or actuators.”Materials Letters. Vol.45 ,pp.70-80,2000.
22.K. Koga and H. Ohigashi,“Piezoelectricity and related properties of vinylidene fluoride and trifluoroethylene copolymers. ”J. Appl. Phys., Vol59,pp. 2142-2150, March 1986.
23.C. K. Lee,“Laminated piezopolymer plates for torsion and bending sensors and actuators.”J. acoust. Soc. Am. Vol.85, pp. 2432-2439,1989.
24.E.F. Crawley, and J. De Luis “Use of piezoelectric actuators as elements of intelligent structures.”AIAA Journal. Vol.25, No.10, pp. 1373-1385,1987
25.X. Y. Ye, Z. y. Zhou, Y. Yang, J. H. Zhang, and J. Yao,“Determination of the mechanical properties of microstructures.”Sensors and Actuators. A54, pp. 750-754,1996.
26.J. G. Smits, and W.S. Choi,“The Constituent Equations of Piezoelectric Heterogeneous Bimorphs.”IEEE Trans. Ultranson. Frequency Contr. Vol.38, no. 3,pp.256-270,1991.
27.S. Brooks, and P. Heyligers, “Static behavior of piezoelectric laminates with distributed and patched actuators.”Journal of Intelligent Material systems and structures. Vol. 5,pp. 635-646,1994.
28.Y.K. Kang, H. C. Park, W. Hwang, and K. S. Han, “Optimum placement of piezoelectric sensor/actuator for vibration control of laminated beams.”AIAA Journal, Vol.30, No.3, pp.347-357,1992.
29.A. Baz, “Boundary control of beams using active constrained layer damping .”ASME. Vol.119, pp.166-172,Apr 1997.