Predicting the type of nonlinearity of shallow spherical shells: Comparison of direct normal form with modal derivatives

Nonlinear vibrations of free-edge shallow spherical shells with large amplitudes are investigated, with the aim of predicting the type of nonlinearity (hardening/softening behaviour) for each mode of the shell, as a function of the radius R of curvature of the shell, from the plate case (R →∞) to the limit of non-shallow shell. Two different models (based on von Kármán’s assumptions or on full numerical finite element approach), and two different methods (normal form and modal derivatives) are contrasted.