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Nonlinear Vibrations of Simply Supported Cylindrical Panels with Uncertain Parameters: An Intrusive Application of the Generalized Polynomial Chaos Expansion

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Abstract

Purpose

This work investigates a simply supported cylindrical panel (with uncertainties in its thickness, radius and Young’s modulus) subjected to a time-dependent transverse load.

Methods

The nonlinear equilibrium equations of the panel are obtained from Donnell’s shallow shell theory, in terms of the transverse displacement field and Airy’s stress function. To discretize this set of equations, the standard Galerkin method is applied in the panel’s space domain. For that, a consistent modal solution for the transverse displacement field is chosen from a perturbation method, considering the main modal couplings that occur due to the geometric nonlinearities present in the equilibrium equations. The uncertainties of the parameters are assumed to have a uniform probability density function and are considered in the nonlinear equilibrium equations by a generalized polynomial chaos expansion of the modal amplitudes of the transverse displacement field. Legendre-chaos polynomials are used to describe the random parameters and the stochastic Galerkin method is applied to discretize the expanded nonlinear equilibrium equations in the random parameter space.

Results

The mean and variance of the bifurcation diagrams are obtained, evaluating the influence of uncertain parameters on their paths, and the regions, where an intrusive application of a generalized polynomial chaos expansion procedure can be applied.

Conclusions

The numerical results show good agreement between the proposed procedure and the results obtained by the Monte Carlo method in the bifurcation diagrams, especially those regions that do not have dynamic jumps and/or competition between the periodic, quasi-periodic and chaotic permanent response.

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Acknowledgements

This work was made possible by the support of the Brazilian agencies: CNPq and FAPEG. Anna Elizabete F. Palla acknowledges the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), which part-financed her studies.

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Authors and Affiliations

  1. School of Civil and Environmental Engineering, Federal University of Goiás, UFG, Av. Universitária, no. 1488, Setor Universitário, Goiânia, GO, CEP: 74605-220, Brazil

    Anna Elizabete F. Palla & Frederico M. A. Silva

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  1. Anna Elizabete F. Palla
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  2. Frederico M. A. Silva
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Correspondence to Frederico M. A. Silva.

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Palla, A.E.F., Silva, F.M.A. Nonlinear Vibrations of Simply Supported Cylindrical Panels with Uncertain Parameters: An Intrusive Application of the Generalized Polynomial Chaos Expansion. J. Vib. Eng. Technol. 10, 2917–2934 (2022). https://doi.org/10.1007/s42417-022-00527-7

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