Abstract
Cylindrical shells exhibit a dense frequency spectrum, especially near the lowest frequency range. In addition, due to the circumferential symmetry, frequencies occur in pairs. So, in the vicinity of the lowest natural frequencies, several equal or nearly equal frequencies may occur, leading to a complex dynamic behavior. So, the aim of the present work is to investigate the dynamic behavior and stability of cylindrical shells under axial forcing with multiple equal or nearly equal natural frequencies. The shell is modelled using the Donnell nonlinear shallow shell theory and the discretized equations of motion are obtained by applying the Galerkin method. For this, a modal solution that takes into account the modal interaction among the relevant modes and the influence of their companion modes (modes with rotational symmetry), which satisfies the boundary and continuity conditions of the shell, is derived. Special attention is given to the 1:1:1:1 internal resonance (four interacting modes). Solving numerically the governing equations of motion and using several tools of nonlinear dynamics, a detailed parametric analysis is conducted to clarify the influence of the internal resonances on the bifurcations, stability boundaries, nonlinear vibration modes and basins of attraction of the structure.
Highlights
Cylindrical shells exhibit a dense frequency spectrum, especially near the lowest frequency range
Due to the circumferential symmetry, frequencies occur in pairs
No consistent modal solution taking into account the simultaneous effect of modal coupling plus modal interaction is found in literature [3, 4]
Summary
Cylindrical shells exhibit a dense frequency spectrum, especially near the lowest frequency range. The aim of the present work is (a) to deduce a consistent modal solution for a shell where two modes exhibits equal natural frequencies, retaining all modes that appear due to the nonlinear modal coupling and modal interaction, (b) obtain from this general expression a precise low order model for large amplitude vibrations, and (c) use this modal solution to study the nonlinear oscillations and instabilities of these cylindrical shells under axial loads This modal solution is obtained by applying perturbation techniques, as suggested previously by the authors in [5, 6]
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