部份嵌入壓電材料懸臂樑，以將壓電材料用部份嵌入結構中的方式，使得整體結構更近似完整的直樑結構，可以避免結構突起所會造成的外在影響，並採用有限元素法來探討壓電複合樑的動態響應且用模態法來驗證有限元素法之可行性；以此有限元素方式做回授控制，探討此一結構在回授控制的抑制振動效果。

在此嵌入式直樑結構分為三個跨矩討論，皆為 Timoshenko 理論的三明治樑所組成。第一三跨距雖三層都為鋁材結構可當成單層結構來看，但在計算上，將之當成三層結構對於整體結構的計算，邊界條件的帶入則會更為方便。

在模態法方面，為了解壓電層樑之力學行為，則利用應力場、應變場與位移的關係推導出應變能項和動能項，再以 Hamilton’s Principle求得壓電層樑之運動方程式，進而計算出模態頻率，並討論在不同幾何參數下對模態頻率的影響。

在有限元素法方面，則使用靜態結構方程式推導出其位移場之通解，再借由應變能項與動能項計算出結構的勁度矩陣和質量矩陣，建立出有限元素模型，利用堆疊方式經 Lagrange’s equation 解出系統的模態頻率，且選取不同數目之元素來堆疊此結構，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 數值積分法對此結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。

In this paper, Finite element method and analytic method is employed to study vibration control of Timoshenko beam with partly embedded piezoelectric material. To compare with the modal frequencies calculated by the two methods, the finite element solutions are approached to the analytic solutions.
In vibration analysis and vibration control, the actuator provide a damping by coupling a negative velocity feedback control algorithm in a closed control loop. Use Newmark method to compute the dynamic response of entire beam.
The results of gain effects, embedded displace effects and embedded length effects can good increse the suppression of vibration.

1. G. R. Cowper, "The shear coefficient in Timoshenko's beam theory," American Society of Mechanical Engineers, Journal of Applied Mechanics 33, 335-340 (1966)
2. D. H. Robbins and J. N. Reddy, "Analysis of piezoelectrically actuated beams using a layer-wise displacement theory," Computers and Structures 41(2), 265-279 (1991)
3. J. G. Smits and W. Choi, "The constituent equations of piezoelectric heterogeneous bimorphs," Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on 38(3), 256-270 (1991)
4. C. K. Lee, "Piezoelectric laminates: theory and experiments for distributed sensors and actuators," Intelligent Structural Systems 77-167 (1992)
5. S. Brooks and P. Heyliger, "Static Behavior of Piezoelectric Laminates with Distributed and Patched Actuators," Journal of Intelligent Material Systems and Structures 5(5), 635-646 (1994)
6. S. M. Yang and Y. J. Lee, "Interaction of structure vibration and piezoelectric actuation," Smart Materials and Structures 3(4), 494-500 (1994)
7. D. A. Saravanos and P. R. Heyliger, "Coupled Layerwise Analysis of Composite Beams with Embedded Piezoelectric Sensors and Actuators," Journal of Intelligent Material Systems and Structures 6(3), 350-363 (1995)
8. M. D. Sciuva and U. Icardi, "Large deflection of adaptive multilayered Timoshenko beams," Composite structures 31(1), 49-60 (1995)
9. X. D. Zhang and C. T. Sun, "Formulation of an adaptive sandwich beam," Smart Materials and Structures 5(6), 814-823 (1996)
10. S. H. Chen, Z. D. Wang and X. H. Liu, "Active Vibration Control and Suppression for Intelligent Structures," Journal of Sound and Vibration 200, 167-177 (1997)
11. J. A. Rongong, J. R. Wright, R. J. Wynne and G. R. Tomlinson, "Modelling of a hybrid constrained layer/piezoceramic approach to active damping," Journal of vibration and acoustics 119(1), 120-130 (1997)
12. I. Y. Shen, "A variational formulation, a work-energy relation and damping mechanisms of active constrained layer treatments," Journal of vibration and acoustics 119(2), 192-199 (1997)
13. X. Q. Peng, K. Y. Lam and G. R. Liu, "Active Vibration Control of Composite Beams with Piezoelectrics: A Finite Element Model with Third Order Theory," Journal of Sound and Vibration 209, 635-650 (1998)
14. B. J. G. Vautier and S. O. R. Moheimani, "Avoiding hysteresis in vibration control using piezoelectric laminates," in Decision and Control, Proceedings. 42nd IEEE Conference on, pp. 269-274 Vol.261 (2003).
15. Y.-H. Zhou and J. Wang, "Vibration control of piezoelectric beam-type plates with geometrically nonlinear deformation," International Journal of Non-Linear Mechanics 39(6), 909-920 (2004)
16. S. Raja, R. Sreedeep and G. Prathap, "Bending Behavior of Hybrid-Actuated Piezoelectric Sandwich Beams," Journal of Intelligent Material Systems and Structures 15(8), 611-619 (2004)
17. Y. Nam, J. Joh, M. Sasaki and H. Yamada, "Vibration suppression of a piezoelectric beam using a self-sensing algorithm," in ICMIT 2005: Information Systems and Signal Processing, pp. 60412Y-60416, SPIE (2005).
18. M. N. Ghasemi-Nejhad, S. Pourjalali, M. Uyema and A. Yousefpour, "Finite Element Method for Active Vibration Suppression of Smart Composite Structures using Piezoelectric Materials," Journal of Thermoplastic Composite Materials 19(3), 309-352 (2006)
19. C. N. Della and D. Shu, "Vibration of beams with piezoelectric inclusions," International Journal of Solids and Structures 44(7-8), 2509-2522 (2007)