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Global dynamics and integrity of a two-dof model of a parametrically excited cylindrical shell

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Abstract

In this paper the global dynamics and topological integrity of the basins of attraction of a parametrically excited cylindrical shell are investigated through a two-degree-of-freedom reduced order model. This model, as shown in previous authors’ works, is capable of describing qualitatively the complex nonlinear static and dynamic buckling behavior of the shell. The discretized model is obtained by employing Donnell shallow shell theory and the Galerkin method. The shell is subjected to an axial static pre-loading and then to a harmonic axial load. When the static load is between the buckling load and the minimum post-critical load, a three potential well is obtained. Under these circumstances the shell may exhibit pre- and post-buckling solutions confined to each of the potential wells as well as large cross-well motions. The aim of the paper is to analyze in a systematic way the bifurcation sequences arising from each of the three stable static solutions, obtaining in this way the parametric instability and escape boundaries. The global dynamics of the system is analyzed through the evolution of the various basins of attraction in the four-dimensional phase space. The concepts of safe basin and integrity measures quantifying its magnitude are used to obtain the erosion profile of the various solutions. A detailed parametric analysis shows how the basins of the various solutions interfere with each other and how this influences the integrity measures. Special attention is dedicated to the topological integrity of the various solutions confined to the pre-buckling well. This allows one to evaluate the safety and dynamic integrity of the mechanical system. Two characteristic cases, one associated with a sub-critical parametric bifurcation and another with a super-critical parametric bifurcation, are considered in the analysis.

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Author notes

  1. Frederico M. A. Silva

    Present address: Civil Engineering Department, Federal University of Goiás, UFG, 74605-220, Goânia, GO, Brazil

Authors and Affiliations

  1. Civil Engineering Department, Pontifical Catholic University, PUC-Rio, 22451-900, Rio de Janeiro, RJ, Brazil

    Paulo B. Gonçalves & Frederico M. A. Silva

  2. Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza”, Via A. Gramsci 53, 00197, Rome, Italy

    Giuseppe Rega

  3. Dipartimento di Architettura, Costruzioni e Strutture, Università Politecnica delle Marche, Via Brecce Bianche, 60131, Ancona, Italy

    Stefano Lenci

Authors
  1. Paulo B. Gonçalves
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  2. Frederico M. A. Silva
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  3. Giuseppe Rega
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  4. Stefano Lenci
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Correspondence to Paulo B. Gonçalves.

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Gonçalves, P.B., Silva, F.M.A., Rega, G. et al. Global dynamics and integrity of a two-dof model of a parametrically excited cylindrical shell. Nonlinear Dyn 63, 61–82 (2011). https://doi.org/10.1007/s11071-010-9785-4

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  • DOI: https://doi.org/10.1007/s11071-010-9785-4

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