本文將採用模態法與有限元素法來探討嵌入式壓電複合層曲樑的模態頻率；此結構之第一和第三跨距為同材料三層的Timoshenko曲樑，而第二跨距為三層的三明治壓電複合層曲樑所組成。為瞭解壓電曲樑之力學行為，則利用應力場、應變場、電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曲樑之運動方程式。
在模態法方面利用位移場與應力場之關係計算出模態頻率。在有限元素法方面，則擷取三層同材料與壓電複合層的一個元素，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，再藉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與模態法求出之解析解作比較，並討論在不同的幾何結構下對模態頻率之影響。
在回授控制方面，利用有限元素法搭配動態阻尼方式，經Newmark’s數值積分法對此節構進行動態回授控制模擬其抑制振動的制振情況。並探討壓電嵌入的方向不同、Gain值效應、壓電材料嵌入位置、和壓電材料嵌入長度對於整體抑制振動的效果影響。

This study presents the natural frequencies and vibration suppression of curved sandwich beam partly embedded with piezoelectric material by Finite Element Method.Base on Timoshinko beam theory, the shape functions of the entire beam are obtained by solving the equations static equilibrium. The modal frequencies obtained from finite element method are compared with those of analytic method. One piezoelectric layer is used as sensor electrode, another as actuator to provide a dampimg by coupling a negative velocity feed back control algorithm in a closed loop.
Newmark method is used in computing the dynamic response of entire curved beam. Further, both effects of location and length of the actuators/sensors on the vibration suppression of the beam also are investigated

1. A. E. H. Love,“A Treatist on the Mathematical Theory of Elasticity”, 4thEdn. Dover,New York, 1944.
2. J. P. Den Hartog,“Mecahanical Vibrations, 4th Edn. MaGraw-Hill”, New York, 1956.
3. E. Volerra and J. D. Morrel,“Lowest natural frequency of elastic hinged arcs”, Journal of the Acoustical Society of America, Vol 33, pp.1787-1790,1961
4. S. S. Rao,“Effects of transverse shear and rotary inertia on the coupled twist-bending vibrations of circular rings”, Journal of Sound amd Vibration, Vol.16, pp.551-556, 1971
5. D. E. Panayotounakos and P. S. Theocaris,“The dynamically loaded circular beam on an elastic foundation”, Journal of Applied Mechanics, Vol 47, pp.139-144, 1980
6. Silva, Julio M.M and Urgueira, Antonio P. V.,“Out-of-plane dynamic response of curved beams-an analytical model”, International Journal of Solid and Structures, Vol.24, No.3, pp.271-284, 1988
7. G. E. Martin,“Determination of equivalent-circuit constants of piezoelectric resonators of moderately low Q by absolute-admittance measurements”, Journal of the Acoustic Society of America, Vol.26, No.3, pp. 413-420, May 1954.
8. 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.
9. 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.
10. 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.
11. 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.
12. 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.
13. 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.
14. 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.
15. You-He Zhou, and Jizeng Wang,“Vibration control of piezoelectric beam-type plates with geometrically nonlinear deformation”, International Journal of Non-Linear Mechanics Vol.39, pp. 909-920, 2004.
16. Christian N Della and Dongwei Shu,“Vibration of beams with embedded piezoelectric sensors and actuators”, Smart Material and Structures Vol.15 ,pp. 529–537, 2006.
17. T. Bailey, & J. E. Hubbard Jr,“Distributed piezoelectric-polymer active vibration control of a cantilever beam”, Journal of Guidance, Control, and Dynamics, Vol. 8(5), pp.605-611, 1985.
18. 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.
19. Seyoung. Im, and S. N. Atluri,“Effects of a piezo-actuator on a finitely deformed beam subjected to general loading”, AIAA Journal, Vol.27, pp. 1801-1807, 1989.
20. 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.
21. S. H. Chen, Z. D. Wang and X. H. Liu,“Active Vibration Control and Suppression for Intelligent Structures”, Journal of Sound and Vibration pp. 167-177, 1997
22. Y. H. Lim,“Finite-element simulation of closed loop vibration control of a smart plate under transient loading”, Smart Material and Structures, pp. 272-286, 2003.
23. B.J.G. Vautier and S.O.R. Moheimani,“Avoiding hysteresis in vibration control using piezoelectric laminates.”, Decision and Control, Proceedings of the 42nd IEEE Conference on, Vol.1, pp.269-274, 2003.
24. Neil D. Sims, Phillip V. Bayly and Keith A. Young,“Piezoelectric sensors and actuators for milling tool stability lobes”, Journal of Sound and Vibration, Vol. 281, pp.743-762, 2005.
25. Y. C. Fung,“Foundations of Solid Mechanics”, Prentice-Hall, Englewood Cliffs, NJ,. 1965.
26. X. D. Zhang and C. T. Sun,“Formulation of an adaptive sandwitch beam”, Smart Materials and Structures Vol.5(6), pp. 814-823, 1996
27. S. Raja, R. Sreedeep and G. Prathap,“Bending Behavior of Hybrid-actuated Piezoelectric Sandwich Beams”, Journal of Intelligent Material Systems and Structures, Vol. 15, pp.611-619, 2004.