Dynamic Stability of Beams with Piezoelectric Layers Located on a Continuous Elastic Foundation A. R. Nezamabadi, and M. Karami Khorramabadi 1 Abstract—This paper studies dynamic stability of homogeneous beams with piezoelectric layers subjected to periodic axial compressive load that is simply supported at both ends lies on a continuous elastic foundation. The displacement field of beam is assumed based on Bernoulli-Euler beam theory. Applying the Hamilton's principle, the governing dynamic equation is established. The influences of applied voltage, foundation coefficient and piezoelectric thickness on the unstable regions are presented. To investigate the accuracy of the present analysis, a compression study is carried out with a known data. Keywords—Dynamic stability, Homogeneous graded beam- Piezoelectric layer, Harmonic balance method. I. INTRODUCTION HE dynamic stability of structures is a subject of considerable engineering importance, and many investigations have been carried out in this subject. In 1985, Bailey and Hubbard [1] investigated the active vibration control of a cantilever beam using distributed piezoelectric polymer as an actuator. Crawley and de Luis [2] developed analytical models for the dynamic response of a cantilever beam with segmented piezoelectric actuators that are either bonded to an elastic substructure or embedded in a laminated composite. Shen [3] used the finite element method to study the free vibration problems of beams containing piezoelectric sensors and actuators. Pierre and Dowell [4] reported the dynamic instability of plates using an extended incremental harmonic balance method. Liu et al. [5] used a finite element model to analyze the shape control and active vibration suppression of laminated composite plates with integrated piezoelectric sensors and actuators. By a feedback control loop, Tzou and Tseng [6] and Ha et al. [7] formulated three-dimensional incompatible finite elements for vibration control of structures containing piezoelectric actuators and sensors. The dynamic instability of a structure subjected to periodic axial compressive forces has attracted a lot of attention.. Bolotin [8] summarized the results achieved in comprehensive studies for the dynamic stability of machine components and structural members. Briseghella et al. [9] used beam elements without A. R. Nezamabadi, is with the Department of Mechanical Engineering, Islamic Azad University, Arak Branch, Arak, Iran (e-mail: alireza.nezamabadi@gmail.com). M. Karami Khorramabadi is with the Department of Mechanical Engineering, Islamic Azad University, Khorramabad Branch, Khorramabad, Iran (corresponding author to provide phone: +98 661 2204463; fax: +98 661 2214281; e-mail: mehdi_karami2001@yahoo.com). axial deformability to solve the dynamic stability problem of beam structures. The load bending contribution was taken into account by means of a second-order approach. Takahashi et al. [10] investigated dynamically unstable regions of cantilever rectangular plates. They presented the numerical results obtained for various loading conditions that are applied along the edge. Recently, Zhu et al. [11] presented a three- dimensional theoretical analysis of the dynamic instability region of functionally graded piezoelectric circular cylindrical shells subjected to a combined loading of periodic axial compression and electric field in the radial direction. To the author's knowledge, there is no analytical solution available in the open literatures for dynamic stability of homogeneous beams with piezoelectric actuators located on a continuous elastic foundation. In the present work, the dynamic stability of a homogeneous beam with piezoelectric actuators subjected to periodic axial compressive loads located on a continuous elastic foundation is studied. Appling the Hamilton's principle, the dynamic equation of beam is derived and solved using the harmonic balance method. The effect of the applied voltages, piezoelectric thicknesses and foundation coefficient on the unstable regions of beam are also discussed. II. FORMULATION Fig. 1 Schematic of the problem studied. Consider a homogeneous beam with piezoelectric actuators and rectangular cross-section as shown in Fig. 1. The thickness, length, and width of the beam are denoted, respectively, by , , L h and . b Also, T h and B h are the thickness of top and bottom of piezoelectric actuators, respectively. The y x − plane coincides with the midplane of the beam and the − z axis located along the thickness direction. The Young's modulus E and the Poisson's ratio ν are assumed to be constant. The beam is assumed to be slender, thus, the Euler-Bernoulli beam theory is adopted. The piezoelectric layers are also assumed to be polarized along the thickness direction. The axial stress and electrical displacement can be written as: T P(t) Piezoelectric Layers x z P(t) World Academy of Science, Engineering and Technology International Journal of Mechanical and Mechatronics Engineering Vol:4, No:2, 2010 202 International Scholarly and Scientific Research & Innovation 4(2) 2010 scholar.waset.org/1307-6892/2350 International Science Index, Mechanical and Mechatronics Engineering Vol:4, No:2, 2010 waset.org/Publication/2350