### A New Approach to the Study of Transverse Vibrations of a Rectangular Plate Having a Circular Central Hole

Document Type: Research Paper

Authors

Department of Mechanical Engineering, University of Kashan

Abstract

In this study, the analysis of transverse vibrations of rectangular plate with circular central hole with different boundary conditions is studied and the natural frequencies and natural modes of a rectangular plate with circular hole have been obtained. To solve the problem, it is necessary to use both Cartesian and polar coordinate system. The complexity of the method is to apply an appropriate model, which can solve the problem of transverse vibrations of a plate. So, it has been tried that the functions of the deflection of plate, in the form of polynomial functions proportionate with finite degrees, to be replaced by Bessel function, which is used in the analysis of the vibrations of a circular plate. Then with the help of a semi-analytical method and orthogonality properties of the eliminated position angle, without any need to analyze so many points on the edges of the rectangular plate, we can prevent the coefficients matrix from becoming so much large as well as the equations from becoming complicated. The above mentioned functions will lead to reducing the calculation time and simplifying the equations as well as speeding up the convergence.

Keywords

[1] Monahan L.J., Nemergut P.J., Maddux G.E., 1970, Natural frequencies and mode shapes of plates with interior cut-outs, The Shock and Vibration Bulletin 41:37-49.
[2] Paramasivam P., 1973, Free vibration of square plates with square opening, Journal of Sound and Vibration 30:173-178.
[3] Aksu G., Ali R., 1976, Determination of dynamic characteristics of rectangular plates with cut-outs using a finite difference formulation, Journal of Sound and Vibration 44:147-158.
[4] Rajamani A., Prabhakaran R., 1977, Dynamic response of composite plates with cut-outs, Part I: Simply-Supported Plates, Journal of Sound and Vibration 54:549-564.
[5] Rajamani A., Prabhakaran R., 1977, Dynamic response of composite plates with cut-outs, Part II: Clamped-Clamped Plates, Journal of Sound and Vibration 54:565-576.
[6] Ali R., Atwal S.J., 1980, Prediction of natural frequencies of vibration of rectangular plates with rectangular cutouts, Computers and Structures 12( 9):819-823.
[7] Lam K.Y., Hung K.C, Chow S.T, 1989, Vibration analysis of plates with cut-outs by the modified rayleigh-ritz method, Applied Acoustics 28:49-60.
[8] Lam K.Y., Hung K.C., 1990, Vibration study on plates with stiffened openings using orthogonal polynomials and partitioning method, Computers and Structures 37:295-301.
[9] Laura P.A., Romanelli E., Rossi R.E., 1997, Transverse vibrations of simply-supported rectangular plates with rectangular cutouts, Journal of Sound and Vibration 202(2):275-283.
[10] Sakiyama T., Huang M., Matsuda H., Morita C., 2003, Free vibration of orthotropic square plates with a square hole, Journal of Sound and Vibration 259(1):63-80.
[11] Joga-Rao C.V., Pickett G., 1961, Vibrations of plates of irregular shapes and plates with holes, Journal of the Aeronautical Society of India 13(3):83-88.
[12] Kumai T., 1952, The flexural vibrations of a square plate with a central circular hole, Proceedings of the Second Japan National Congress for Applied Mechanics .
[13] Hegarty R.F., Ariman T., 1975, Elasto-dynamic analysis of rectangular plates with circular holes, International Journal of Solids and Structures 11:895-906.
[14] Eastep F.E., Hemmig F.G., 1978, Estimation of fundamental frequency of non-circular plates with free, circular cutouts, Journal of Sound and Vibration 56(2):155-165.
[15] Nagaya K., 1952, Transverse vibration of a plate having an eccentric inner boundary, Journal of Applied Mechanics 18(3):1031-1036.
[16] Nagaya K., 1980, Transverse vibration of a rectangular plate with an eccentric circular inner boundary, International Journal of Solids and Structures 16:1007-1016.
[17] Lee H.S., Kim K.C., 1984, Transverse vibration of rectangular plates having an inner cutout in water, Journal of the Society of Naval Architects of Korea 21(1):21-34.
[18] Kim K.C., Han S.Y., Jung J.H., 1987, Transverse vibration of stiffened rectangular plates having an inner cutout, Journal of the Society of Naval Architects of Korea 24(3):35-42.
[19] Avalos D.R., Laura P.A., 2003, Transverse vibrations of simply supported rectangular plates with two rectangular cutouts, Journal of Sound and Vibration 267:967-977.
[20] Lee H.S., Kim K.C., 1984, Transverse vibration of rectangular plates having an inner cutout in water, Journal of the Society of Naval Architects of Korea 21(1):21-34.
[21] Khurasia H.B., Rawtani S., 1978, Vibration analysis of circular plates with eccentric hole, Journal of Applied Mechanics 45(1):215-217.
[22] Lin W.H., 1982, Free transverse vibrations of uniform circular plates and membranes with eccentric holes, Journal of Sound and Vibration 81(3):425-433.
[23] Laura P.A., Masia U., Avalos D.R., 2006, Small amplitude, transverse vibrations of circular plates elastically restrained against rotation with an eccentric circular perforation with a free edge, Journal of Sound and Vibration 292:1004-1010.
[24] Cheng L., Li Y.Y., Yam L.H., 2003, Vibration analysis of annular-like plates, Journal of Sound and Vibration 262: 1153-1170.
[25] Lee W.M., Chen J.T, Lee Y.T., 2007, Free vibration analysis of circular plates with multiple circular holes using indirect BIEMs, Journal of Sound and Vibration 304:811-830.
[26] Zhong H., Yu T., 2007, Flexural vibration analysis of an eccentric annular mindlin plate, Archive of Applied Mechanics 77:185-195.
[27] Ventsel E., Krauthammer T., 2001, Thin Plates and Shells: Theory, Analysis, and Applications, Marcel Dekker, New York.
[28] Nagaya K., 1980, Transverse vibration of a rectangular plate with an eccentric circular inner boundary, International Journal of Solids and Structures 16:1008-1016.