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IssuesArchive of Issues2013-2pp.186-193

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K.V. Avramov and I.D. Breslavskii, "Vibrations of Shallow Shells Rectangular in the Horizontal Projection with Two Freely Supported Opposite Edges," Mech. Solids. 48 (2), 186-193 (2013)
Year 2013 Volume 48 Number 2 Pages 186-193
DOI 10.3103/S0025654413020106
Title Vibrations of Shallow Shells Rectangular in the Horizontal Projection with Two Freely Supported Opposite Edges
Author(s) K.V. Avramov (Podgorny Institute for Problems in Mechanical Engineering, National Academy of Sciences of Ukraine, Dm. Pozharskogo 2/10, Kharkiv, 61046 Ukraine, kvavr@kharkov.ua)
I.D. Breslavskii (National Technical University "Kharkiv Polytechnic Institute," Frunze 21, Kharkiv, 61002 Ukraine, id.breslavsky@gmail.com)
Abstract The exact mode shapes of linear vibrations of a shallow shell rectangular in the horizontal projection with two freely supported opposite edges are obtained. These shapes are used to construct a discrete model of vibrations of a shallow shell in geometrically nonlinear deformation. The harmonic balance method is used to study the free and forced nonlinear vibrations under internal resonance. The Lyapunov stability of the obtained periodic vibrations is analyzed.
Keywords shallow shell, nonlinear vibrations, internal resonance, stability of motion
References
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2.  A. W. Leissa and A. S. Kadi, "Curvature Effects on Shallow Shell Vibrations," J. Sound Vibr., 16 (2), 173-187 (1971).
3.  A. S. Vol'mir, A. A. Logvinskaya, and V. V. Rogalevich, "Nonlinear Free Vibrations of Rectangular Plates and Cylindrical Panels," in Proc. 8th All-Union Conf. on Theory of Shells and Plates (Nauka, Moscow, 1973), pp. 426-431 [in Russian].
4.  M. Amabili, "Nonlinear Vibrations of Doubly Curves Shallow Shells," Int. J. Nonlin. Mech. 40 (5), 683-710 (2005).
5.  A. W. Leissa, "The Free Vibration of Rectangular Plates," J. Sound Vibr. 31 (3), 257-293 (1973).
6.  L. Zhang and Y. Xiang, "Vibration of Open Circular Cylindrical Shells with Intermediate Ring Supports," Int. J. Solids Struct. 43 (13), 3705-3722 (2006).
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10.  M. Amabili, Nonlinear Vibrations and Stability of Shells and Plates (Cambridge Univ. Press, Cambridge, 2008) [in Russian].
11.  K. Avramov, Yu. Mikhlin, and E. Kurilov, "Asymptotical Analysis of Nonlinear Dynamics of Simply Supported Cylindrical Shells," Nonlin. Dyn. 47 (4), 331-352 (2007).
12.  I. Breslavskii, K. V. Avramov, Yu. Mikhlin, and R. Kochurov, "Nonlinear Modes of Snap-Through Motions of a Shallow Arch," J. Sound Vibr. 311 (1-2), 297-313 (2008).
13.  V. S. Anishchenko, Complex Vibrations in Simple Systems (Nauka, Moscow, 1990) [in Russian].
14.  V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients and Their Applications (Nauka, Moscow, 1972) [in Russian].
Received 26 April 2010
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