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Mathematical Problems in Engineering
Volume 2006 (2006), Article ID 35672, 25 pages

Dynamics and statics of flexible axially symmetric shallow shells

1Department of Automatics and Biomechanics, Technical University of Lodz, 1/15 Stefanowskiego Street, Lodz 90924, Poland
2Department of Mathematics, Saratov State University, Saratov 410054, Russia

Received 24 June 2005; Accepted 12 July 2005

Copyright © 2006 J. Awrejcewicz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In this work, we propose the method for the investigation of stochastic vibrations of deterministic mechanical systems represented by axially symmetric spherical shells. These structure members are widely used as sensitive elements of pressure measuring devices in various branches of measuring and control industry, machine design, and so forth. The proposed method can be easily extended for the investigation of shallow spherical shells, goffer-type membranes, and so on. The so-called charts of control parameters for a shell subjected to a transversal uniformly distributed and local harmonic loading force and resistance moment are constructed. The scenarios of the transition of vibration of shallow-type system into chaotic state are investigated with the use of the theory of differential equations and the theory of nonlinear dynamics. The method of the control of chaotic vibrations of flexible spherical shells subjected to a transversal harmonic load through a synchronized action of either harmonic resistance moment or force is proposed, illustrated, and discussed.