本文目的為探討一根單邊貼附有壓電片之 懸臂樑及簡支樑上承受移動負載之振動分析，採用模態法來探討壓電懸臂樑及簡支樑的動態響應。
在模態法方面，為瞭解壓電樑的力學行為，則利用應力場、應變場與位移的關係於推導應變能與動能，再以 求得壓電樑之運動方程式，求取此樑的模態形狀函數與模態頻率，並討論在不同幾何參數對模態頻率的影響。
在求得出模態形狀函數及頻率後，運用模態形狀函數及模態疊加法推導出動態方程式，於此樑承受移動負載作用力之研究並使用 數值分析法求解動態方程式，進而探討樑的位移以及壓電片的電荷收集情形，最後加上電阻後，探討此樑之各種幾何因素對於抑制此樑的震動之影響進行探討。

The purpose of this thesis is to study the dynamic responses of moving load on cantilever beam and simply-supported beam, which has a piezoelectric sheet mounted on the bottom surface. The Timoshenko theory is adopted in this study. The displacement and rotation of all components of the entire beam are set. The displacements, stresses, strains, electric field and electric displacements are used to derive electrical enthalpy, strain energy and kinetic energy of the entire beam. The governing equations and the corresponding boundary condition are derived via the Hamilton’s principle. The natural frequencies are obtained by analytical method. Dynamic analysis is based on the method of modal analysis. The method is presented to obtain the dynamic responses of the entire beam induced by a load moving on the beam. Results show that the traveling velocity of the load and the geometric parameters of the beam significantly affect both histories of the displacement of the beam and the electric charge accumulation on the piezoelectric surface. Furthermore, the effect of vibration suppression via implementing a resistor connected both surfaces of the piezoelectric sheet will also be investigated.

參考文獻
[1] M. Di Sciuva and U. Icardi, "Large deflection of adaptive multilayered Timoshenko beams," Composite structures, vol. 31, pp. 49-60, 1995.
[2] E. F. Crawley and J. De Luis, "Use of piezoelectric actuators as elements of intelligent structures," AIAA journal, vol. 25, pp. 1373-1385, 1987.
[3] J. Yang, Y. Chen, Y. Xiang, and X. Jia, "Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load," Journal of Sound and Vibration, vol. 312, pp. 166-181, 2008.
[4] G. Lewis and F. Monasa, "Large deflections of cantilever beams of nonlinear materials," Computers & Structures, vol. 14, pp. 357-360, 1981.
[5] C. Sun and X. Zhang, "Use of thickness-shear mode in adaptive sandwich structures," Smart Materials and Structures, vol. 4, p. 202, 1995.
[6] D. Robbins and J. Reddy, "Analysis of piezoelectrically actuated beams using a layer-wise displacement theory," Computers & structures, vol. 41, pp. 265-279, 1991.
[7] A. Erturk, "Piezoelectric energy harvesting for civil infrastructure system applications: moving loads and surface strain fluctuations," Journal of Intelligent Material systems and structures, vol. 22, pp. 1959-1973, 2011.
[8] G. E. Martin, "Determination of Equivalent‐Circuit Constants of Piezoelectric Resonators of Moderately Low Q by Absolute‐Admittance Measurements," The Journal of the Acoustical Society of America, vol. 26, pp. 413-420, 1954.
[9] G. Bradfield, "Ultrasonic transducers: 1. Introduction to ultrasonic transducers Part A," Ultrasonics, vol. 8, pp. 112-123, 1970.
[10] A. Triplett and D. D. Quinn, "The effect of non-linear piezoelectric coupling on vibration-based energy harvesting," Journal of Intelligent Material Systems and Structures, vol. 20, pp. 1959-1967, 2009.
[11] J. G. Smits and W.-s. Choi, "The constituent equations of piezoelectric heterogeneous bimorphs," IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 38, pp. 256-270, 1991.
[12] Y.-H. Zhou and J. Wang, "Vibration control of piezoelectric beam-type plates with geometrically nonlinear deformation," International Journal of Non-Linear Mechanics, vol. 39, pp. 909-920, 2004.
[13] N. N. Rogacheva, The theory of piezoelectric shells and plates: CRC Press, 1994.
[14] 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.
[15] J. H. Huang and H.-I. Yu, "Dynamic electromechanical response of piezoelectric plates as sensors or actuators," Materials Letters, vol. 46, pp. 70-80, 2000.
[16] S. A. Siddiqui, M. F. Golnaraghi, and G. R. Heppler, "Dynamics of a flexible cantilever beam carrying a moving mass," Nonlinear Dynamics, vol. 15, pp. 137-154, 1998.
[17] R. Katz, C. W. Lee, A. G. Ulsoy, "The dynamic response of a rotating shaft subject to a moving load," Journal of Sound and Vibration, vol. 122, pp. 131-148, 1988.
[18] J.-D. Yau, "Vibration of simply supported compound beams to moving loads," Journal of Marine Science and Technology, vol. 12, pp. 319-328, 2004.
[19] J. Pan, C. H. Hansen, and S. D. Snyder, "A study of the response of a simply supported beam to excitation by a piezoelectric actuator," Journal of Intelligent Material Systems and Structures, vol. 3, pp. 3-16, 1992.
[20] N. Liu and G.-L. Yang, "Vibration property analysis of axially moving cantilever beam considering the effect of moving mass," Journal of Vibration and Shock, vol. 3, pp.102-105, 2012.
[21] T. Yoshimura, J. Hino and N. Ananthanarayana, “Vibration analysis of nonlinear beam subjected to moving loads by use Galerkin method,” Journal of Sound and Vibration, vol. 104, pp. 179-186, 1986.
[22] M. Olsson, “On the fundamental moving load problem,”
Journal of Sound and Vibration, vol. 145, pp. 299-307, 1991.
[23] 吳弘志,具單層貼附式壓電材料之Timoshenko樑振動分析,
成功大學工程科學系碩士論文, 2013.

[24] Zhu, Shining, Bei Jiang, and Wenwu Cao. "Characterization of piezoelectric materials using ultrasonic and resonant techniques." Medical Imaging'98. International Society for Optics and Photonics, 1998.
[25] Raja, S., R. Sreedeep, and Gangan Prathap. "Bending behavior of hybrid-actuated piezoelectric sandwich beams." Journal of Intelligent Material Systems and Structures 15.8 (2004): 611-619.