Abstract
An analytical solution of the problem of free in-plane vibration of rectangular plates with complicated boundary conditions is proposed.
An analytical solution of the problem of free in-plane vibration of rectangular plates with complicated boundary conditions is proposed.
I. V. Andrianov and J. Awrejcewicz, “New trends in asymptotic approaches: summation and interpolation methods,” Applied Mechanics Reviews, vol. 54, no. 1, pp. 69–92, 2001.
View at: Google ScholarI. V. Andrianov, V. Z. Gristchak, and A. O. Ivankov, “New asymptotic method for the natural, free and forced oscillations of rectangular plates with mixed boundary conditions,” Technische Mechanik, vol. 14, no. 3-4, pp. 185–193, 1994.
View at: Google ScholarI. V. Andrianov and A. O. Ivankov, “Application of Padé approximants in the method of introducing a parameter in the investigation of biharmonic equations with complex boundary conditions,” USSR Computational Mathematics and Mathematical Physics, vol. 27, no. 1, pp. 193–196, 1987.
View at: Google ScholarI. V. Andrianov and A. O. Ivankov, “New asymptotic method for solving of mixed boundary value problem,” in Free Boundary Problems in Continuum Mechanics (Novosibirsk, 1991), vol. 106 of Internat. Ser. Numer. Math., pp. 39–45, Birkhäuser, Basel, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNetI. V. Andrianov and A. O. Ivankov, “On the solution of the plate bending mixed problems using modified technique of boundary conditions perturbation,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 73, no. 2, pp. 120–122, 1993.
View at: Google Scholar | Zentralblatt MATHJ. Awrejcewicz, I. V. Andrianov, and L. I. Manevitch, Asymptotic Approaches in Nonlinear Dynamics. New Trends and Applications, Springer Series in Synergetics, Springer, Berlin, 1998.
View at: Zentralblatt MATH | MathSciNetG. A. Baker and P. Graves-Morris, Padé Approximants, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2nd edition, 1996.
View at: Zentralblatt MATH | MathSciNetN. S. Bardell, R. S. Langley, and J. M. Dunsdon, “On the free in-plane vibration of isotropic rectangular plates,” Journal of Sound and Vibration, vol. 191, no. 3, pp. 459–467, 1996.
View at: Publisher Site | Google ScholarA. N. Bercin and R. S. Langley, “Application of the dynamical stiffness technique to the in-plane vibration of plate structures,” Computers & Structures, vol. 59, no. 5, pp. 869–875, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATHA. A. Dorodnitzyn, “Using of small parameter method for numerical solution of mathematical physics equations,” Numerical Methods for Solving of Continuum Mechanics Problems (Collection of Works), VZ AN SSSR, Moscow, pp. 85–101, 1969 (Russian).
View at: Google ScholarR. El Mokhtari, J.-M. Cadou, and M. Potier-Ferry, “A two grid algorithm based on perturbation and homotopy method,” Comptes Rendus Mecanique, vol. 330, no. 12, pp. 825–830, 2002.
View at: Publisher Site | Google ScholarA. Elhage-Hussein, M. Potier-Ferry, and N. Damil, “A numerical continuation method based on Padé approximants,” International Journal of Solids and Structures, vol. 37, no. 46-47, pp. 6981–7001, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetN. H. Farag and J. Pan, “Free and forced in-plane vibration of rectangular plates,” Journal of the Acoustical Society of America, vol. 103, no. 1, pp. 408–413, 1998.
View at: Publisher Site | Google ScholarN. H. Farag and J. Pan, “Modal characteristics of in-plane vibration of rectangular plates,” Journal of the Acoustical Society of America, vol. 105, no. 6, pp. 3295–3310, 1999.
View at: Publisher Site | Google ScholarD. J. Gorman, “Accurate analytical type solution for the free in-plane vibration of clamped and simply supported rectangular plates,” Journal of Sound and Vibration, vol. 276, no. 1-2, pp. 311–333, 2004.
View at: Publisher Site | Google ScholarD. J. Gorman, “Free in-plane vibration analysis of rectangular plates by the method of superposition,” Journal of Sound and Vibration, vol. 272, no. 3–5, pp. 831–851, 2004.
View at: Publisher Site | Google ScholarD. J. Gorman, “Free in-plane vibration analysis of rectangular plates with elastic support normal to the boundaries,” Journal of Sound and Vibration, vol. 285, no. 4-5, pp. 941–966, 2005.
View at: Publisher Site | Google ScholarD. J. Gorman, “Exact solutions for the free in-plane vibration of rectangular plates with two opposite edges simply supported,” Journal of Sound and Vibration, vol. 294, no. 1-2, pp. 131–161, 2006.
View at: Publisher Site | Google ScholarK. F. Graff, Wave Motion in Elastic Solids, Dover, New York, 1991.
R. H. Gutierezz and P. A. A. Laura, “In-plane vibrations of thin, elastic, rectangular plates elastically restrained against translation along the edges,” Journal of Sound and Vibration, vol. 132, no. 3, pp. 512–515, 1989.
View at: Publisher Site | Google ScholarJ. L. Guyander, C. Boisson, and C. Lesueur, “Energy transmission in finite coupled plates, part1: theory,” Journal of Sound and Vibration, vol. 81, no. 1, pp. 81–92, 1982.
View at: Publisher Site | Google Scholar | Zentralblatt MATHK. Hyde, J. Y. Chang, C. Bacca, and J. A. Wickert, “Parameter studies for plane stresses in-plane vibration of rectangular plates,” Journal of Sound and Vibration, vol. 247, no. 3, pp. 471–487, 2001.
View at: Publisher Site | Google ScholarR. S. Langley and A. N. Bercin, “Wave intensity analysis of high frequency vibrations,” Philosophical Transactions: Physical Sciences and Engineering, vol. 346, no. 1681, pp. 489–499, 1994.
View at: Google ScholarS. Liao, Beyond perturbation. Introduction to the Homotopy Analysis Method, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Florida, 2004.
View at: Zentralblatt MATH | MathSciNetR. H. Lyon, “In-plane contribution to structural noise transmission,” Noise Control Engineering Journal, vol. 26, pp. 22–27, 1985.
View at: Google ScholarS. G. Mihlin, Variational Methods in Mathematical Physics, Pergamon Press, Oxford, New York, 1964.
View at: Zentralblatt MATHF. Molenkamp, J. B. Sellmeijer, C. B. Sharma, and E. B. Lewis, “Explanation of locking of four-node plane element by considering it as elastic Dirichlet-type boundary value problem,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 24, no. 13, pp. 1013–1048, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATHB. Ovunc, “In-plane vibration of plates by continuous mass matrix method,” Computers & Structures, vol. 8, no. 6, pp. 723–731, 1978.
View at: Publisher Site | Google Scholar | Zentralblatt MATHJ. Seok and H. F. Tiersten, “Free vibration of annular sector cantilever plates. Part 2: in-plane motion,” Journal of Sound and Vibration, vol. 271, no. 3–5, pp. 773–787, 2004.
View at: Publisher Site | Google ScholarJ. Seok, H. F. Tiersten, and H. A. Scarton, “Free vibration of rectangular cantilever plates. Part 2: in-plane motion,” Journal of Sound and Vibration, vol. 271, no. 1-2, pp. 147–158, 2004.
View at: Publisher Site | Google ScholarA. V. Singh and T. Muhammad, “Free in-plane vibration of isotropic non-rectangular plates,” Journal of Sound and Vibration, vol. 273, no. 1-2, pp. 219–231, 2004.
View at: Publisher Site | Google ScholarG. Wang, S. Veeramani, and N. M. Wereley, “Analysis of sandwich plates with isotropic face plates and a viscoelastic core,” Journal of Vibration and Acoustics, vol. 122, no. 3, pp. 305–312, 2000.
View at: Publisher Site | Google ScholarG. Wang and N. M. Wereley, “Free in-plane vibration of rectangular plates,” AIAA Journal, vol. 40, no. 5, pp. 953–959, 2002.
View at: Google Scholar