Non-linear vibrations of imperfect free-edge circular plates and shells

Large-amplitude, geometrically non-linear vibrations of free-edge circular plates with geometric imperfections are addressed in this work. The dynamic analog of the von Kármán equations for thin plates, with a stress-free initial deflection, is used to derive the imperfect plate equations of motion. An expansion onto the eigenmode basis of the perfect plate allows discretization of the equations of motion. The associated non-linear coupling coefficients for the imperfect plate with an arbitrary shape are analytically expressed as functions of the cubic coefficients of a perfect plate. The convergence of the numerical solutions are systematically addressed by comparisons with other models obtained for specific imperfections, showing that the method is accurate to handle shallow shells, which can be viewed as imperfect plate. Finally, comparisons with a real shell are shown, showing good agreement on eigenfrequencies and mode shapes. Frequency-response curves in the non-linear range are compared in a very peculiar regime displayed by the shell with a 1:1:2 internal resonance. An important improvement is obtained compared to a perfect spherical shell model, however some discrepancies subsist and are discussed.