Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 678307, 19 pages
Research Article

Effect of Imperfections and Damping on the Type of Nonlinearity of Circular Plates and Shallow Spherical Shells

Cyril Touzé,1 Cédric Camier,1 Gaël Favraud,1 and Olivier Thomas2

1ENSTA-UME, Unité de Mécanique, Chemin de la Hunière, 91761 Palaiseau Cedex, France
2CNAM-LMSSC, Laboratoire de Mécanique des Structures et Systèmes Couplés, 2 rue Conté, 75003 Paris, France

Received 28 November 2007; Accepted 20 February 2008

Academic Editor: Paulo Gonçalves

Copyright © 2008 Cyril Touzé et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The effect of geometric imperfections and viscous damping on the type of nonlinearity (i.e., the hardening or softening behaviour) of circular plates and shallow spherical shells with free edge is here investigated. The Von Kármán large-deflection theory is used to derive the continuous models. Then, nonlinear normal modes (NNMs) are used for predicting with accuracy the coefficient, the sign of which determines the hardening or softening behaviour of the structure. The effect of geometric imperfections, unavoidable in real systems, is studied by adding a static initial component in the deflection of a circular plate. Axisymmetric as well as asymmetric imperfections are investigated, and their effect on the type of nonlinearity of the modes of an imperfect plate is documented. Transitions from hardening to softening behaviour are predicted quantitatively for imperfections having the shapes of eigenmodes of a perfect plate. The role of 2:1 internal resonance in this process is underlined. When damping is included in the calculation, it is found that the softening behaviour is generally favoured, but its effect remains limited.