Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von Kármán's theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.