Welcome to Open Science
Vibrations Analysis of Circular Plate with Piezoelectric Actuator Using Thin Plate Theory and Bessel Function
Current Issue
Volume 2, 2015
Issue 6 (November)
Pages: 140-156   |   Vol. 2, No. 6, November 2015   |   Follow on
Paper in PDF Downloads: 98   Since Oct. 14, 2015 Views: 1561   Since Oct. 14, 2015
Authors
[1]
[2]
Abstract
Vibrations of flexible structures have been an important engineering study owing to its both deprecating and complimentary traits. These flexible structures are generally modeled as strings, bars, shafts and beams (one dimensional), membranes and plate (two dimensional) or shell (three dimensional). Structures in many engineering applications, such as building floors, aircraft wings, automobile hoods or pressure vessel end-caps, can be modeled as plates. Undesirable vibrations of any of these engineering structures can lead to catastrophic results. It is important to know the fundamental frequencies of these structures in response to simple or complex excitations or boundary conditions. In this study vibration of thin plates is discussed using both analytical and approximate methods. The method of Boundary Characteristic Orthogonal Polynomials (BCOP) is presented which helps greatly in simplifying computational analysis. First of all it eliminates the need of using trigonometric and Bessel functions as admissible functions for the Raleigh Ritz analysis and the Assumed Mode Method. It produces diagonal or identity mass matrices that help tremendously in reducing the computational effort. The BCOPs can be used for variety of geometries including rectangular, triangular, circular and elliptical plates. The boundary conditions of the problems are taken care of by a simple change in the first approximating function. Using these polynomials as admissible functions, frequency parameters for circular and annular plates are found to be accurate up to fourth decimal point. A simplified model for piezoelectric actuators is then derived considering the isotropic properties related to displacement and orthotropic properties of the electric field. The equations of motion for plate with patch are derived using equilibrium (Newtonian) approach as well as extended Hamilton’s principle. The solution of equations of motion is given using BCOPs and fundamental frequencies are then found. In the final, the experimental verification of the plate vibration frequencies is performed with electromagnetic inertial actuator and piezoelectric actuator using both circular and annular plates.
Keywords
Plate, Orthogonal Polynomials, Piezoelectric Actuator
Reference
[1]
Laura, P.A.A. and Sanchez, M.D., Vibrations of circular plates of rectangular orthotropy, Journal of Sound and Vibration, 209(1), 199-202, 1998.
[2]
Pistonesi, C. and Laura, P.A.A., Forced vibrations of a clamped, circular plate of rectangular orthotropy, Journal of Sound and Vibration, 228(3), 712-16, 1999.
[3]
Topalian, V.; Larrondo, H.A.; Avalos, D.R. and Laura, P.A.A., Journal of Sound and Vibration, 202(1), 125-33, 1997.
[4]
Gutierrez, R.H. and Laura, P.A.A., Transverse vibrations of a circular plate of polar anisotropy with a concentric circular support, Journal of Sound and Vibration, 231(4),1175-8, 2000.
[5]
Vera, S.A.; Laura, P.A.A. and Vega, D.A., Transverse vibrations of a free-free circular annular plate, Journal of Sound and Vibration, 224(2), 379-83, 1999.
[6]
Laura, P.A.A. and Gutierrez, R.H., Analysis of vibrating circular plates of nonuniform thickness by the method of differential quadrature, Ocean Engineering, 22(1), 97-100, 1995.
[7]
Wu, T.Y.; Wang, Y.Y. and Liu, G.R., Free vibration analysis of circular plates using generalized differential quadrature rule, Computer Methods in Applied Mechanics and Engineering,191(46), 5365-80, 2002.
[8]
Gupta, U.S.; Lal, R. and Sharma, S., Vibration analysis of non-homogeneous circular plate of nonlinear thickness variation by differential qudrature method, Journal of Sound and Vibration, 298(4-5), 892-906, 2006.
[9]
Chorng-Fuh Liu and Ge-Tzung Chen, A simple finite element analysis of axisymmetric vibration of annular and circular plates, International Journal of Mechanical Sciences, 37(8), 861-71, 1995.
[10]
Lakis, A.A. and Selmane, A., Classical solution shape functions in the finite element analysis of circular and annular plates, International Journal for Numerical Methods in Engineering, 40(6), 969-90, 1997.
[11]
Selmane, A. and Lakis, A.A., Natural frequencies of transverse vibrations of non-uniform circular and annular plates, Journal of Sound and Vibration, 220(2), 225-49, 1999.
[12]
Chorng-Fuh Liu, Ting-Jung Chen and Ying-Jie Chen, A modified axisymmetric finite element for the 3-D vibration analysis of piezoelectric laminated circular and annular plates, Journal of Sound and Vibration, 309(3-5), 794-804,. 2008.
[13]
Rao S.S., Vibrations of Continuous Systems, John Wiley & Sons Inc, New Jersey, 2007. [14] Bingen, Y., Stress, Strain, and Structural Dynamics: An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes, Academic Press, 2005.
[14]
Gray A. and Matthews G.B., A treatise on Bessel functions and their applications to physics, McMillan and Co, New York, 1895.
[15]
Jenkins, C. and Korde U., Membrane vibration experiments: An historical review with recent results, Journal of Sound and Vibration, 295(3-5), 602-613, 2006.
[16]
Gaspar J. and Pappa R., Membrane Vibration experiments using surface bonded piezoelectric patch actuation, NASA/TM-2003-212150.
[17]
Sewall J., Miserantino R. and Pappa R., Vibration studies of a Lightweight Three side membrane suitable for space application, NASA Technical Paper 2095, 1988.
[18]
Leissa, A., Vibration of Plates, NASA SP 160, 1969.
[19]
Gray A. and Matthews G.B., A treatise on Bessel functions and their applications to physics, McMillan and Co, New York, 1895.
[20]
Chakraverty S., Vibration of plates, CRC Press, 2009.
[21]
Bingen, Y., Stress, Strain, and Structural Dynamics: An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes, Academic Press, 2005.
[22]
Bhat R. and Rajalingham C., Axisymmetric Vibration of Circular Plates and its Analog in Elliptical Plates using Characteristic Orthogonal Polynomials, Journal of Sound and Vibrations.161(1),109-118, 1993.
[23]
Ventsel E. and Krauthhammer T., Thin Plates and Shell: Theory Analysis and Applications, New York, Marcel Dekker, 2001.
[24]
Ikeda, T., Fundamentals of Piezoelectricity, Oxford University Press, 1990.
[25]
Banks H. and Smith R., “The modeling of piezo ceramic patch interactions with shells, plates and beams”, ICASE report No 92-66, 1992.
[26]
Lew K.M. and Lam K.Y., “A Raleigh Ritz approach to transverse vibration of isotropic and anisotropic trapezoidal plates using orthogonal plate functions”, International Journal of Solids and Structures, 27(2), 189-203, 1991.
[27]
Singh B. and Saxena V., “Transverse vibration of a quarter of an elliptic plate with variable thickness”, International Journal of Mechanical Sciences, 37(10), 1103-1132, 1995.
[28]
Cui Q., Liu C. and Zha X., “Modeling and numerical analysis of a circular piezoelectric actuator for valve-less micropumps”, Journal of Intelligent Material Systems and Structures, 19(10), 1195-1205, 2008.
[29]
Azimi, S., Free vibration of circular plates with elastic edge supports using the reacceptances method, Journal of Sound and Vibration, 120(1), 19-35, 1987.
[30]
Kim, C and Dickinson, S., On the lateral vibration of thin annular and circular composite plates subject to certain complicating effects, Journal of Sound and Vibration, 130(3),363-377, 1988.
[31]
Kim, C and Dickinson S. The flexible vibration of thin isotropic and polar orthotropic annular and circular plate with elastically restrained peripheries, Journal of Sound and Vibrations, 143(1), 171-179, 1990.
[32]
Jalili, N. and Knowles, D., Structural vibration control using an active resonator absorber: modeling and control implementation, Smart Materials and Structures, 13(5), 998-1005, 2004.
[33]
Olgac, N and Jalili, N., Modal analysis of flexible beams with delayed resonator vibration absorber: Theory and Experiments, Journal of Sound and Vibration, 218(2), 307-331, 1998.
[34]
Vogel, S. and Skinner, D, Natural frequencies of transversely vibrating annular plates, Journal of Applied mechanics, 32,926-931, 1965.
[35]
Narita, Y., Natural frequencies of completely free annular and circular plates having polar orthotropic, Journal of Sound and Vibration, 92(1), 33-38, 1984.
[36]
Material properties of micro fiber composite actuator taken from website: http://www.smart-material.com/Smart-choice.php?from=MFC.
[37]
Jalili, N., “Piezoelectric based vibration control: from macro to micro/Nano scale systems”, Springer, 2009.
[38]
Young, Y., Tang L., and Li, H., Vibration energy harvesting using macro fiber composites, Smart materials and structures, (18), 115025, 2009.
Open Science Scholarly Journals
Open Science is a peer-reviewed platform, the journals of which cover a wide range of academic disciplines and serve the world's research and scholarly communities. Upon acceptance, Open Science Journals will be immediately and permanently free for everyone to read and download.