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

Cyril Touzé (1) , Olivier Thomas (2)

1 DFA - Dynamique des Fluides et Acoustique
2 LMSSC - Laboratoire de Mécanique des Structures et des Systèmes Couplés

Non-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.

Mots clés: Shallow spherical shells; Hardening/softening behaviour; Non-linear normal modes; Internal resonance

Domaines: Génie mécanique [physics.class-ph]; Génie mécanique [physics.class-ph]

Cyril Touzé, Olivier Thomas. Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry. International Journal of Non-Linear Mechanics, 2006, 41 (5), pp.678-692. ⟨10.1016/j.ijnonlinmec.2005.12.004⟩. ⟨hal-00838885⟩