Acoplamento e interação modal em painéis cilíndricos imperfeitos e simplesmente apoiados - Silva, Wanclaine Almeida Vaz da (2019)
Internal resonance is one of the main areas of research in the field of nonlinear dynamics. This phenomenon occurs when the proportion of the natural frequencies of distinct vibration modes is an integer, and the internal resonance allows the exchange of energy between the vibration modes involved, influencing the stability of the dynamic responses. In this work deals the analyses of linear and nonlinear free and forced vibrations of a simply supported cylindrical panel, subjected to a time dependent transverse loading, considering the modal coupling and modal interaction phenomena in the proposed modal solution for the transversal displacement field. The transversal equations of motion are obtained through an energetic approach using the deformation field and mean surface curvature changes given by Donnell's nonlinear theory for shallow shells. The system of nonlinear partial differential equations is determined by minimizing the Lagrangian energy of structural system. From classical vibration modes, an analysis of free vibrations is performed to determine the geometries that generate modal interaction between two or three distinct vibration modes, it is noteworthy that the geometries found in this dissertation have internal resonances of the type: 1: 1, 1: 2, 1: 3, 1: 1: 2 and 1: 2: 4. The proposed modal solution for the transversal displacement field is achieved by employing the perturbation method, considering in the initial modal solution the existing modal interaction between the two or three distinct vibration modes. From the Galerkin method, nonlinear partial differential equations are discretized giving rise to a system of nonlinear ordinary differential equations. The results of nonlinear free vibrations of a perfect cylindrical panel, or with a geometric imperfection in the form of one of the modes of vibration present in the modal solution for transverse displacements, are expressed as a function of the number of degrees of freedom considered in the modal solution the transverse displacement field. It is also made an analyse of the forced vibrations, determining the resonance curves, the basin of attraction, phase planes and mapping the Poincaré section of the perfect and imperfect cylindrical panel. Thus, it can be observed that the shape and magnitude of the initial geometric imperfections have a great influence on the nonlinear responses of both free and forced vibrations.