本文之目的為探討一個單邊嵌入壓電材料Timoshenko beam之動態分析，由有限元素法建立其運動方程模型，以此配合Newmark’s scheme模擬探討此結構受暫態外力與感應電壓之動態行為。
　　本文將懸臂樑結構分為三個跨距，每一跨距具有上下兩層，僅第二跨距結構上層為壓電材料，而第一、三跨距及第二跨距的下層均為鋁材；並由樑結構之位移場、應力場和應變場推導出應變能與動能，且由Hamilton原理求得結構Governing equations及邊界條件。
利用靜態平衡方程式推導出各個跨距的單位元素之位移函數向量，經由單位元素的節點位移以有限元素法分別堆疊出各個跨距的勁度矩陣與質量矩陣，並以Lagrange’s equations建立系統的運動耦合方程式組。
以Newmark’s Scheme模擬樑系統之動態響應情形，分別探討施加外力於結構末端和施加電壓於壓電材料作用，以了解當改變壓電材料長度、位置與厚度時，樑結構的位移變化及壓電片輸出電荷情形。

In this thesis the cantilevered Timoshenko beam with one-side embedded a piezoelectric material is presented. The governing equation is set up by the finite element approach. The dynamics of structure subjected to an external force and applied voltage on the piezoelectric can be investigated by the Newmark’s scheme.

The Timoshenko beam is regarded as a three-span beam and each span has top and bottom components. The upper layer of the second span is a piezoelectric material. All the other parts are Al material. The total electric enthalpy and kinetic energy of the entire beam are obtained with displacements, stresses and strains. The governing equations and the corresponding boundary condition are derived via by Hamilton’s principle.

The displacement vectors of each span element are obtained by solving the equations of static equilibrium. Then, the finite element approach is adopted to pile up the stiffness and mass matrixes of each span from element’s nodal displacements. The coupling equations of the motion of the system can then be set up via the Lagrange’s equations.

The Newmark’s scheme is adopted to compute the dynamic responses of the entire system. The force acting at the tip of the entire beam and a voltage applied on the piezoelectric material are taken as two examples. The effects of length, location and thickness of the piezoelectric material on the history of displacement of the beam and the history of the charge accumulation on the piezoelectric surfaces are investigated.

1.C. K. Lee, “Theory of laminated piezoelectric plates for the design of distributed sensors/actuators. Part I: governing equations and reciprocal relationship,” J. Acoust. Soc. Am., Vol.87, No.3, pp. 1144-1158, 1990.
2.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.
3.D. T. Detwiler, M.-H.H. Shen and V. B. Venkayya, “Finite element analysis of laminated composite structures containing distributed piezoelectric actuators and sensors,” Finite Elements in Analysis and Design, Vol. 20, pp. 87-100, 1995.
4.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, Vol. 209, No. 4, pp. 635-650, 1998.
5.G.R. Liu, X.Q. Peng, K. Y. Lam and J. Tani, “Vibration control simulation of laminated composite plates with integrated piezoelectrics,” Journal of Sound and Vibration, Vol. 220, No. 5, pp. 827-846, 1999.
6.I.Y. Shen, “A variation formulation, a work-energy relation and damping mechanisms of active constrained layer treatments,” Journal of Vibration and Acoustics, Vol. 119, pp. 192-199, 1997.
7.M.A. Trindade, A. Benjeddou and R. Ohayon, “Piezoelectric active vibration control of damped sandwich beams,” Journal of Sound and Vibration, Vol. 246, No. 4, pp. 653-677, 2001.
8.J. H. Hung and H. I. Yu, “Dynamic electromechanical response of piezoelectric plates as sensors or actuators,” Materials Letters, Vol. 46, pp. 70-80, 2000.
9.M. C. Ray, K. M. Rao and B. Samanta, “Exact solution for static analysis of an intelligent structure under cylindrical bending,” Computers and Structures, Vol. 47, No. 6, pp. 1031-1042, 1993.
10.H. Abramovich and A. Livshits, “Dynamic behavior of cross-ply laminated beams with piezoelectric layers,” Composite Structures, Vol. 25, pp. 371-379, 1993.
11.M. D. Sciuva and U. Icardi, “Large deflection of adaptive multilayered Timoshenko beams,” Composite Structures , Vol. 31, pp. 49-60, 1995.
12.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.