5 Conclusions and future work In the present study, the problem of arbitrary-shaped vibrating Kirchhoff plates was analyzed using the Rayleigh-Ritz method. General boundary conditions were introduced in the functional of potential energy, and natural coordinates were employed to express the geometry of plates of arbitrary shape in a simple form. Flexural free vibration analysis of different shaped plates was performed, showing the effectiveness of the method. The flexural frequencies obtained were compared with those found in the literature and with those of standard finite element analysis, and the results were in very good agreement. Future studies will be devoted to analyzing the effectiveness of varying the composition of a given set of such functions. Acknowledgments This study was developed within the INTERMECH laboratory with the contribution of the Regione Emilia Romagna - Assessorato Attività Produttive, Sviluppo Economico, Piano telematico, PRRIITT misura 3.4 azione A Obiettivo 2. References [1] Leissa A.W., “Recent research in plate vibrations: classical theory”, The Shock and Vibration Digest 9 (10), pp. 1324, 1977. [2] Durvasula S., “Natural frequencies and modes of clamped skew plates”, American Institute of Aeronautics and Astronautics Journal 7, pp. 1164-1167, 1969. [3] Babu P.V.T., Reddy D.V., “Frequency analysis of skew orthotropic plates by the finite strip method”, Journal of Sound and Vibration 18 (4-5), pp. 465-474, 1971. [4] Ramakrishnan R., Kunukkasseril V., “Free vibration of annular sector plates”, Journal of Sound and Vibration 30, pp. 127-129, 1973. [5] Li W.Y., Cheung Y.K., Tham L.G., “Spline finite strip analysis of general plates”, Journal of Engineering Mechanics 112, pp. 43-54, 1986. [6] Mizusawa. T., Kajita. T., Naruoka. M., “Analysis of skew plate problems with various constraints”, Journal of Sound and Vibration 73 (4), pp. 575-584, 1980. [7] Geannakakes G., “Vibration analysis of arbitrarily shaped plates using beam characteristic orthogonal polynomials in the semi-analytical finite strip method”, Journal of Sound and Vibration 137 (2), pp. 283-303, 1990. [8] Ritz W., “Theorie der transversalschwingungen einer quadratischen Platte mit freien Rändern”, Ann. Physik 28, pp. 737786, 1909. [9] Catania G., Sorrentino S., “Rayleigh-Ritz analysis of vibrating plates based on a class of eigenfunctions”. In proceedings of ASME IDETC/CIE, San Diego, USA, August 30 - September 2, 2009. [10] Catania G., Sorrentino S., “Discrete spectral modelling of continuous structures with fractional derivative viscoelastic behaviour”. In proceedings of ASME IDETC/CIE, Las Vegas, USA, September 4-7, 2007. [11] Catania G., Fasana A., Sorrentino S., “A condensation technique for the FE dynamic analysis with fractional derivative viscoelastic models”, Journal of Vibration and Control 14 (9-10), pp. 1573-1586, 2008. [12] Timoshenko S., Young D.H., Weaver W., Vibration problems in engineering, 4th edition, Wiley, New York, USA, 1974. [13] Cook R.D., Malkus D.S., Plesha M.E., Witt R.J., Concepts and applications of finite element analysis, 4th edition, Wiley, New York, USA, 2002. [14] Blevins R.D., Formulas for natural frequency and mode shape, Krieger, Malabar, USA, 1979-2001. [15] Chakraverty S., Vibration of plates, CRC Press, New York, USA, 2009. [16] Leissa A.W., Vibration of Plates, Nasa SP 160, U.S. Government Printing Office, Washington D.C., USA, 1969. 88
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