https://hal-ensta-paris.archives-ouvertes.fr/hal-00838885Touzé, CyrilCyrilTouzéDFA - Dynamique des Fluides et Acoustique - UME - Unité de Mécanique - ENSTA Paris - École Nationale Supérieure de Techniques AvancéesThomas, OlivierOlivierThomasLMSSC - Laboratoire de Mécanique des Structures et des Systèmes Couplés - CNAM - Conservatoire National des Arts et Métiers [CNAM] - HESAM - HESAM Université - Communauté d'universités et d'établissements Hautes écoles Sorbonne Arts et métiers universitéNon-linear behaviour of free-edge shallow spherical shells: Effect of the geometryHAL CCSD2006Shallow spherical shellsHardening/softening behaviourNon-linear normal modesInternal resonance[PHYS.MECA.GEME] Physics [physics]/Mechanics [physics]/Mechanical engineering [physics.class-ph][SPI.MECA.GEME] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Mechanical engineering [physics.class-ph]Arnoux, Aurélien2016-03-18 17:32:102022-08-05 14:53:592016-03-21 09:44:30enJournal articleshttps://hal-ensta-paris.archives-ouvertes.fr/hal-00838885/document10.1016/j.ijnonlinmec.2005.12.004application/x-download1Non-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.