本文將採用模態法與有限元素法來探討壓電複合層樑的模態頻率；此結構中的第一和第三跨距為單層的Timoshenko樑，第二跨距為三層的三明治壓電複合層樑所組成。
在模態法方面，為瞭解壓電複合層樑之力學行為，則利用應力場、應變場與位移場的關係推導出應變能項和動能項，再以漢米爾頓原理求得壓電複合層樑之運動方程式，利用位移場與應力場之關係計算出模態頻率，並討論在不同的幾何參數下對模態頻率之影響。
在有限元素法方面，擷取單層與三明治層的一個元素，並且以靜態平衡模式找出此元素各節點位移與轉角之形狀函數，計算此塊有限元素的表示式，再借由應變能項與動能項計算出此結構的勁度矩陣和質量矩陣，進而利用Lagrange’s equation及堆疊技巧解出系統的模態頻率，並將結果與模態法之結果作比較，進一步確定有限元素法的可行性。
在加入適當的電壓找出其壓電三明治樑結構整體之位移，最後利用限元素法搭配開迴路回授設計方式，作為制抑此結構之振動。

In this thesis, the finite element technique is developed for the vibration analysis and vibration control of Timoshenko beam. The beam structure has one segment of piezoelectric sandwich beam. The displacement fields are set up. The strains, stresses, stress resultants and stress-couple resultants, kinetic energy and electrical enthalpy of the entire beam are derived. The governing equations are formulated via the Hamilton’s principle.
The shape function of the entire beam is obtained by solving the equations static equilibrium. Then, the technique of finite element is employed to compute the modal frequencies of the entire beam.The modal frequencies obtained from finite element computation and analytic method, respectively, will be compared to show the feasibility of finite element computation. Then, the direct piezoelectric beam equation is used to calculate the total charge on the sensor electrode, and the actuator provides a damping by coupling a negative velocity feedback control algorithm in a closed control loop.
Newmark method is used in computing the dynamic response of entire beam. Further, the efficiency of the both location and electric current of the actuators/sensors on the vibration control system of the beam also is investigated.

1.G. E. Martin, “Derermination of equivalent-circuit constants of piezoelectric resonators of moderately low Q by absolute-admittance measurements.” The Journal of the Acoustic Society of America, Vol.26, No.3, pp. 413-420, May 1954.
2.T. B. Bailey, and J. E. Hubbard, “ Distributed piezoelectric-polymer active vibration control of a cantilever beam.” Journal of Guidance Control, and Dynamics, Vol.8, No.5, pp. 605-611, 1985.
3.G. Bradfield, “Ultrasonic transducers 1. Introduction transducers. Part A.” Ultrasonics, pp. 112-123, 1970.
4.H. S. Tzou, and G. C. Wang, “Distributed structural dynamics control of flexible manipulators-I. Structural dynamics and viscoelastic actuator.” Composite and Structures, Vol.35 , pp. 669-667, 1990.
5.S. K. Ha, C. Keilers, and F. K. Chang, “Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators.” AIAA Journal, Vol.35 , pp. 669-677, 1990.
6.H. Abramovich, and A. Livshits, “Dynamic behavior of cross-ply laminated beams with piezoelectric actuators.” Composite Structures. Vol.25 , pp.371-379, 1993.
7.Z. Chaudhry and C. A. Rogers, “Enhanced structural contral with discretely attached induced strain actuators.” Transaction of the ASME, Journal of Mechanical Design, Vol. 115, pp. 718-722, 1993.
8.K. Koga and H. Ohigashi, “Piezoelectricity and related properties of vinylidene fluoride and trifluoroethylene copolymers.” Journal of applied physics, Vol. 59, pp. 2142-2150, March 1986.
9.C. K. Lee, “Laminated piezopolymer plates for torsion and bending sensors and actuators.” Journal of the Acoustical Society of America, Vol. 85, pp. 2432-2439, 1989.
10.C. Q. Chen, X. M. Wang and Y. P. Shen, “Finite element approach of vibration control using self-sensing piezoelectric actuators.” Computers and Structures, Vol.60, No.3, pp. 505-512, 1996.
11.H. S. Tzou, and G. C. Wang, “ Distributed structural dynamics control of flexible manipulators-I. Structural dynamics and viscoelastic actuator.” Computers and Structures, Vol.35, pp. 669-677, 1990.
12.S. K. Ha, C. Keilers, and F. K. Chang, “ Finite element analysis of composite structures containing distributed piezoelectric sensors and actuators.” AIAA Journal, Vol.30, pp. 772-780, 1992.
13.A. Benjeddou, M. A. Trindade, and R. Ohayon, “ New Shear Actuated Smart Structure Beam Finite Element.” AIAA Journal, Vol.37, pp. 378-383, 1999.
14.D. H. Robbins, and J. N. Reddy, “Analysis of piezoelectrically actuated beams using a layer-wise displacement theory.” Computers & Structures, Vol. 41, no. 2. pp. 265-279, 1991.
15.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.
16.X. Y. Ye, Z. Y. Zhou, Y. Yang, J. H. Zhang, and J. Yao, “ Determinaation of the mechanical properties of microstructructures.” Sensors and Actuators, A54, pp. 750-754, 1996.
17.J. G. Smits, and W. S. Choi, “The Constituent Equations of Piezoelectric Heterogeneous Bimorphs.” Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions, Vol. 38, no. 3, pp. 256-270, 1991.
18.J. G. Smits, “Desigen consideration of a Piezoelectric-on-silicon microrobot.” Sensor and Actuators, A35, pp. 129-135, 1992.
19.S. Im, and S. N. Atluri, “Effects of piezo-actuator on a finitely deformed beam subjected to general loading.” AIAA Journal, Vol.27, pp. 1801-1807, 1989.
20.S. Brooks, and P. Heyliger, “Static behavior of piezoelectric laminates with distributed and patched actuators.” Journal of Intelligent Material Systems and Structures, Vol. 5, pp. 635-646, 1994.
21.G. P. Dube, S. Kapuria, and P. C. Dumir, “Exact piezothermoelastic solution of simply-supported orthotropic flat panel in cylindrical bending.” Journal of Intelligent Material Systems and Structures. Vol. 38, no. 11, pp. 1161-1177, 1996.
22.M, Di Sciuva, and U. Icardi, “ Large deflection of adaptive multilayered Timoshenko beams.” Composite Structures, Vol.31, pp. 49-60, 1995.
23.J. L. Dion, E. Cornieles, F. Galindo, and K. Agbossou, “Exact one-dimensional computation of ultrasonic transducers with several piezoelectric elements and passive layers using the transmission line analogy.” Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions, Vol. 44, no. 5, pp. 1120-1129, Sep. 1997.
24.J. A. Mitchell, and J. N. Reddy, “A refined hybrid plate theory for composite laminates with piezoelectric laminae.” International journal of solids and structures, Vol.32, No.16, pp. 2345-2367, 1995.
25.S. Shen, and Z. B. Kuang, “An active control model of laminated piezothermoelastic plate.” International journal of solids and structures, Vol.36, pp. 1925-1945, 1999.
26.D. A. Saravannos, P. R. Heyliger, and D. A. Hopkins, “Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates.” International journal of solids and structure,. Vol.34, No.3, pp. 359-378, 1997.
27.Q. Wang and S. T. Quek, “A model for the analysis of beams with embedded piezoelectric layer.” Journal of Intelligent Material Systems and Structures, Vol.13, pp. 61-70, 2002.
28.R. L. Goldberg, M. J. Jurgents, D. M. Mills, C. S. Henrique, D. Vaughan, and S. W. Smith, “Modeling of piezoelectric multilayer ceramics using finite element analysis.” Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions, Vol. 44, no. 6, pp. 1204-1210, Nov. 1997.
29.Q. Meng, M. J. Jurgents, and K. Deng, “ Modeling of the electromechanical performance of piezoelectric laminated microactuators.” Journal of micromechanics and microengineering, Vol. 3, pp. 18-23, 1993.
30.H. J. Ding, and W. Q. Chen, “New state space formulations for transversely isotropic piezoelectricity with applications.” Mechanics research communications, Vol.27, No.3, pp. 319-326, 2000.
31.Xu K, Noor A. K., and Tang Y. Y. , “Three-dimensional solutions for free vibrations of initially-stressed thermoelectroelastic multilayered plates.” Computer Methods in Applied Mechanics and Engineering, Vol.141, pp. 125-139, 1997.
32.D. A. Saravanos, and P. R. Heyliger, “Coupled layerwise analysis of composite beams with embedded piezoelectric sensor and actuators.” Journal of Intelligent Material Systems and Structures, Vol. 6, pp. 350-369, 1995.
33.J. G. Smits, and A. Ballato, “Dynamic admittance matrix of cantilever bimorphs.” Journal of Microelectromech System, Vol. 3, no. 3, pp. 105-112, 1994.
34.Jin H. Huang, and Heng-I Yu, “Dynamic electromechanical response of piezoelectric plates as sensors or actuators.” Materials Letters, Vol.45 , pp. 70-80, 2000.
35.J. G. Smits, W. S. Choi, and A. Ballato, “Resonance and antiresonance of symmetric and asymmetric piezoelectric flexors.” Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions, Vol. 44, no. 2, pp. 250-258, 1997.
36.J. Pan, C. H. Hansen, and S. D. Snyder, “A study of response of a simple supported beam to excitation by a piezoelectric actuator.” Journal of Intelligent Material Systems and Structures, Vol.3, pp.120-130, 1992..
37.M. C. Ray, K. M. Rao, and B. Samanta, “Exact solution for static analysis of an intelligent structure under cylindrical bending.” Computer and Structures, Vol. 47,no. 6, pp. 1031-1042, 1993.
38.D. Certon, O. Casula, F. Patat, and D. Royer, “Theoretical and experimental investigations of lateral modes in 1-3 piezocompsites.” Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions, Vol. 44, no. 3, pp. 643-651, March 1997.
39.H. Abramovich, and A. Livshits, “Dynamic behavior of cross-ply laminated beams with piezoelectric layers.” Computer Structures, Vol. 25, pp. 371-379, 1993.
40.G. S. Agnes, “Development of a model for simultaneous active and passive piezoelectric vibration suppression.” Journal of Intelligent Material Systems and Structures, Vol. 6, pp. 482-487, 1995.
41.D. X. Liao, C. K. Sung, and B. S. Thompson, “ The optimum design of symmetric laminated beams considering damping.” Journal of Composite Materials, Vol.20, pp.489-501, 1986.
42.A. Baz, “Boundary control of beams using active constrained layer damping.” Transaction of the ASME, Journal of Vibration and Acoustics, Vol. 119, pp. 166-172, 1997.
43.H. S. Tzou, and C. I. Tseng, “Distributed vibration control and Identification of Coupled Elastic/Piezoelectric System: Finite Element Formulatino and Application.” Mechanical Systems and Signal Processing, Vol.5, No.3, pp. 215-231, 1991.
44.J. A. Rongong, J. R. Wright, R. J. Wynne, and G. R. Tomlinson, “Modeling of a hybrid constrained laryer/piezoceramic approach to active damping.” Transaction of the ASME, Journal of Vibration and Acoustics, Vol. 119, pp. 120-130, 1997.
45.S. H. Chen, Z. D. Wang, and X. H. Liu, “Active Vibration Control and Suppression for Intelligent Structurs.” Journal of Sound and Vibration, Vol.200, No.2 , pp. 167-177.1995
46.I. Y. Shen, “A variational formulation, a work-energy relation and damping mechanisms of active constrained layer treatments.” Transaction of the ASME, Journal of Vibration and Acoustics, Vol. 119, pp. 192-199, 1997.
47.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.
48.M. Krommer, and H. Irschik, “A Reissner-Mindlin-type plate theory including the direct piezoelectric and the pyroelectric effect.” Acta Mechanica, Vol.141, pp. 51-69, 2000.
49.G. R. Cowper, “The shear coefficient in Timoshenko’s beam theory.” Journal of Applied Mechanics, Vol.33, pp.335-340, 1966.
50.X. D. Zhang and C. T. Sun “Formulation of an adaptive sandwich beam.” Smart Materials and Structures. Vo5, pp. 814-823, 1996.
51.C. K. Lee, “Piezoelectric laminates: theory and experiment for distribted sensors and actuators.” Intelligent Structural Systems, pp. 75-167, 1992.