Abstract
This paper deals with the free transverse vibration of anisotropic plates with several arbitrarily located internal line hinges and piecewise smooth boundaries, elastically restrained against rotation and translation. The equations of motion and the associated boundary and transition conditions are derived using Hamilton’s principle in a rigorous framework. A new analytical manipulation based on a condensed notation is used to compact the corresponding analytical expressions. A combination of the Ritz method and the Lagrange multipliers method with polynomials as coordinate functions is used to obtain tables of the nondimensional frequencies and the corresponding mode shapes, for rectangular plates with different boundary conditions and restraint conditions in the internal line hinges. The cases not previously treated of two- and three line hinges are particularly analyzed.
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References
Courant R., Hilbert D.: Methods of Mathematical Physics. Interscience Publ., New York (1953)
Guelfand I., Fomin S.: Calculus of Variations. Prentice Hall, Englewood Cliffs (1963)
Kantorovich L., Krylov V.: Approximate Methods of Higher Analysis. Interscience Publishers, New York (1964)
Mikhlin S.: Variational Methods of Mathematical Physics. Mac Millan, New York (1964)
Sagan H.: Introduction to the Calculus of Variations. Mc Graw Hill Book Co., New York (1969)
Bliss G.: Calculus of Variations. The Mathematical Association of América. The Open Court Publ. Co., Chicago (1971)
Weinstock R.: Calculus of Variations with Applications to Physics and Engineering. Dover Publications, New York (1974)
Ewing G.M.: Calculus of Variations with Applications. Dover Publications, New York (1985)
Leitmann G.: The Calculus of Variations and Optimal Control. Plenum Press, New York (1986)
Brechtken-Manderscheid U.: Introduction to the Calculus of Variations. Chapman and Hall, London (1991)
Blanchard P., Brüning E.: Variational Methods in Mathematical Physics. Springer, New York (1992)
Giaquinta M., Hildebrandt S.: Calculus of Variations I. Springer, Berlin (1996)
Troutman J.L.: Variational Calculus and Optimal Control. Springer, New York (1996)
Grossi R.O.: Cálculo de Variaciones. Teoría y aplicaciones. Centro Internacional de Métodos Numéricos en Ingeniería (CIMNE). Editorial Dsignum, Barcelona (2010)
Dym C., Shames I.: Solid Mechanics: A Variational Approach. Mc Graw Hill Book Co., New York (1973)
Szilard R.: Theory and Analysis of Plates. Prentice Hall, Englewood Cliffs (1974)
Timoshenko S., Krieger W.: Theory of Plates and Shells. Mc Graw Hill, New York (1959)
Lekhnitskii S.G.: Anisotropic Plates. Gordon and Breach Science, New York (1968)
Whitney J.M.: Structural Analysis of Laminated Anisotropic Plates. Technomic Publishing Co. Inc., Pennsylvania (1987)
Reddy J.N.: Mechanics of Laminated Anisotropic Plates: Theory and Analysis. CRC Press, Boca Raton (1997)
Jones R.M.: Mechanics of Composite Materials. Taylor and Francis, Philadelphia (1999)
Wang, C.M., Xiang, Y., Wang, C.Y.: Buckling and vibration of plates with an internal line hinge via the Ritz method. In: Proceedings of First Asian-Pacific Congress on Computational Mechanics. Sydney, pp. 1663–1672 (2001)
Gupta P.R., Reddy J.N.: Buckling and vibrations of orthotropic plates with an internal line hinge. Int. J. Struct. Stab. Dyn. 2, 457–486 (2002)
Xiang Y., Reddy J.N.: Natural vibration of rectangular plates with an internal line hinge using the first order shear deformation plate theory. J. Sound Vib. 263, 285–297 (2003)
Huang M., Ma X.Q., Sakiyama T., Matsuda H., Morita C.: Natural vibration study on rectangular plates with a line hinge and various boundary conditions. J. Sound Vib. 322, 227–240 (2009)
Quintana M.V., Grossi R.O.: Free vibrations of a generally restrained rectangular plate with an internal line hinge. Appl. Acoust. 73, 356–365 (2012)
Grossi R.O.: Boundary value problems for anisotropic plates with internal line hinges. Acta Mech. 223, 125–144 (2012)
Grossi R.O.: A Note on the use of variational methods for treatment of plate dynamics. J. Multi Body Dyn. 225, 263–271 (2011)
Grossi R.O.: On the existence of weak solutions in the study of anisotropic plates. J. Sound Vib. 242, 542–552 (2001)
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Grossi, R.O., Raffo, J. Natural vibrations of anisotropic plates with several internal line hinges. Acta Mech 224, 2677–2697 (2013). https://doi.org/10.1007/s00707-013-0892-4
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DOI: https://doi.org/10.1007/s00707-013-0892-4