Acoplamento e interação modal em painéis cilíndricos imperfeitos e simplesmente apoiados
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2019-08-30
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Universidade Federal de Goiás
Resumo
Internal resonance is one of the main areas of research in the field of nonlinear dynamics. This
phenomenon occurs when the proportion of the natural frequencies of distinct vibration modes is
an integer, and the internal resonance allows the exchange of energy between the vibration modes
involved, influencing the stability of the dynamic responses. In this work deals the analyses of
linear and nonlinear free and forced vibrations of a simply supported cylindrical panel, subjected to
a time dependent transverse loading, considering the modal coupling and modal interaction
phenomena in the proposed modal solution for the transversal displacement field. The transversal
equations of motion are obtained through an energetic approach using the deformation field and
mean surface curvature changes given by Donnell's nonlinear theory for shallow shells. The system
of nonlinear partial differential equations is determined by minimizing the Lagrangian energy of
structural system. From classical vibration modes, an analysis of free vibrations is performed to
determine the geometries that generate modal interaction between two or three distinct vibration
modes, it is noteworthy that the geometries found in this dissertation have internal resonances of
the type: 1: 1, 1: 2, 1: 3, 1: 1: 2 and 1: 2: 4. The proposed modal solution for the transversal
displacement field is achieved by employing the perturbation method, considering in the initial
modal solution the existing modal interaction between the two or three distinct vibration modes.
From the Galerkin method, nonlinear partial differential equations are discretized giving rise to a
system of nonlinear ordinary differential equations. The results of nonlinear free vibrations of a
perfect cylindrical panel, or with a geometric imperfection in the form of one of the modes of
vibration present in the modal solution for transverse displacements, are expressed as a function of
the number of degrees of freedom considered in the modal solution the transverse displacement
field. It is also made an analyse of the forced vibrations, determining the resonance curves, the
basin of attraction, phase planes and mapping the Poincaré section of the perfect and imperfect
cylindrical panel. Thus, it can be observed that the shape and magnitude of the initial geometric
imperfections have a great influence on the nonlinear responses of both free and forced vibrations.
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SILVA, W. A. V. Acoplamento e interação modal em painéis cilíndricos imperfeitos e simplesmente apoiados. 2019. 113 f. Dissertação (Mestrado em Geotecnia, Estruturas e Construção Civil) - Universidade Federal de Goiás, Goiânia, 2019.