Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Abstract

International audienceNon-linear vibrations of free-edge shallow spherical shells are investigated, in order to predict the trend of non-linearity (hardening/softening behaviour) for each mode of the shell, as a function of its geometry. The analog for thin shallow shells of von Kármán's theory for large deflection of plates is used. The main difficulty in predicting the trend of non-linearity relies in the truncation used for the analysis of the partial differential equations (PDEs) of motion. Here, non-linear normal modes through real normal form theory are used. This formalism allows deriving the analytical expression of the coefficient governing the trend of non-linearity. The variation of this coefficient with respect to the geometry of the shell (radius of curvature R, thickness h and outer diameter 2 a) is then numerically computed, for axisymmetric as well as asymmetric modes. Plates (obtained as R → ∞) are known to display a hardening behaviour, whereas shells generally behave in a softening way. The transition between these two types of non-linearity is clearly studied, and the specific role of 2:1 internal resonances in this process is clarified. © 2006 Elsevier Ltd. All rights reserved

Non-linear vibrations of free-edge thin spherical shells: Experiments on a 1:1:2 internal resonance

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Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates

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Nonlinear vibrations and chaos in gongs and cymbals

“…When a curvature is present, the type of behaviour (hardening or softening) depends on the balance between quadratic and cubic terms. The transition between these two cases has been studied by Fletcher, for a single mode of simple rods systems, by Thomas et al [13], and, more exhaustively, by Touzé and Thomas for shallow spherical shells [14]. In this latter paper, it is shown that increasing the curvature leads to changing the behaviour from hardening to softening.…”