Nonlinear normal modes to analyze the nonlinear forced vibrations of cylindrical shells with internal resonances

Resumo

In this work, the nonlinear normal modes are applied to analyze the nonlinear forced vibrations of a simply supported cylindrical shell with internal resonances. The nonlinear equilibrium equations are obtained considering the Donnell's nonlinear shallow shell theory. The modal solution to the transversal displacement field, used to discretize the equilibrium equations, is obtained by perturbation techniques that consider an internal resonance between the linear vibration modes. The discretized equations of the reduced order model are obtained using an invariant manifold approach. The nonlinear dynamic behavior is analyzed from the resonance curves that are obtained by the continuation method. The resonance curves are obtained for both full and reduced order models, and these results are compared to determine the level of a harmonic load that can be reliably represented by a reduced order model. Several multi-modes are considered to assemble the best nonlinear normal modes basis that contains the most important information of the interactions that occur between the modes of the transversal displacement field. Time responses and phase portraits, with mapping Poincaré sections, are also used to analyze the nonlinear dynamic behavior of the cylindrical shell and to check the accuracy of the reduced order models.