# Vibration Analysis for Rectangular Plate Having a Circular Central Hole with Point Support by Rayleigh-Ritz Method

Document Type: Research Paper

Authors

Department of Mechanical Engineering, University of Kashan

Abstract

In this paper, the transverse vibrations of rectangular plate with circular central hole have been investigated and the natural frequencies of the mentioned plate with point supported by Rayleigh-Ritz Method have been obtained. In this research, the effect of the hole is taken into account by subtracting the energies of the hole domain from the total energies of the whole plate. To determine the kinetic and potential energies of plate, admissible functions for rectangular plate are considered as beam functions and 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. Consideration for a variety of edge conditions is given through a combination of simply supported, clamped and free boundary conditions. In this study, the effects of increasing the diameter of the hole and the effects of number of point supported on the natural frequencies were investigated and the optimum radius of the circular hole for different boundary conditions are obtained. The method has been verified with many known solutions. Furthermore, the convergence is very fast with any desirable accuracy to exact known natural frequencies.

Keywords

### References

[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, Journal of Sound and Vibration 54:549-564.
[5] Rajamani A., Prabhakaran R., 1977, Dynamic response of composite plates with cut-outs, 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 2nd Japan National Congress on Applied Mechanics 339-342.
[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., , 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] Wang D., Yang Z.C., Yu Z.G.,2010, Minimum stiffness location of point support for control of fundamental natural frequency of rectangular plate by Rayleigh–Ritz method, Journal of Sound and Vibration 329:2792-2808.
[28] Joseph Watkins R., Barton Jr O., 2010, Characterizing the vibration of an elastically point supported rectangular plate using eigensensitivity analysis, Thin-Walled Structures 48:327-333.
[29] Dozio L., 2011, On the use of the trigonometric ritz method for general vibration analysis of rectangular kirchhoff plates, Thin-Walled Structures 49:129-144.
[30] Kwak M.K., Han S.,2007, Free vibration analysis of rectangular plate with a hole by means of independent coordinate coupling method, Journal of Sound and Vibration 306:12-30.
[31] Saeedi K., Leo A., 2012, Vibration of circular plate with multiple eccentric circular perforations by the Rayleigh-Ritz method, Journal of Mechanical Science and Technology 26 (5):1439-1448.
[32] Fan S.C., Cheung Y.K., 1984, Flexural free vibrations of rectangular plates with complex support conditions, Journal of Sound and Vibration 93:81-94.
[33] Utjes J.C., Laura P.A., 1984, Vibrations of thin elastic plates with point supports: a comparative study, Second National Meeting of Users of the Method of Finite Elements.
[34] Wang D., Jiang J.S., Zhang W.H., 2004, Optimization of support positions to maximize the fundamental frequency of structures, International Journal for Numerical Methods in Engineering 61:1584-1602.