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a) = signal
b) = phase diagram
c) = Poincare map
d) = spectrum
e) = deformed shell at the ten times indicated in a).
From:
A.V. Krysko, J. Awrejcewicz, I.V. Papkova, V.M. Zakharov, T.V. Yakovleva and V.A. Krysko,
“Non-linear dynamics of multi-layer shells,
Unidentified chapter in unidentified book in the pdf file, January 2013, DOI: 10.13140/2.1.2092.7043
ABSTRACT: In the first paper part we study the mathematical model of a two-layer cylindrical shell (with clearance between both shells) having constant stiffness and density and subjected to a periodic transversal load action. The derived mathematical model allows us to study non-linear dynamics of both structural members with regard to external load and internal interaction between the two layers, as well as the force action coming from the inside of the second cylinder. The developed mathematical model includes geometric non-linearity of both cylinders and their contact interactions. Differential equations governing the dynamics of both shells are solved by the Bubnov-Galerkin higher order approximation method, whereas the obtained Cauchy problem is solved using the fourth order Runge-Kutta method. Convergence of the Bubnov-Galerkin method versus the number of approximating series terms is considered. Namely, we solve the problem of two embedded cylindrical shells including their interaction as systems with an infinite number of degrees of freedom. In the second paper part chaotic dynamics of continuous multi-layer structural members in the form of flexible shallow spherical panels is analyzed. The PDEs governing panel dynamics are reduced to the Cauchy problem through the FDM (Finite Difference Method) of the second accuracy order, and then the ODEs obtained are analyzed via the fourth-order Runge-Kutta method. A novel scenario of transition from regular to chaotic panel dynamics is reported and illustrated. We develop approaches to detect and monitor spatial chaos, and then we illustrate how both timing and spatial chaos appears simultaneously. We show also how the two-layer structural member exhibits non-symmetric chaotic vibrations, which after transition into windows with periodic dynamics exhibit again symmetric vibrations.
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