Internal resonances in a transversally excited imperfect circular cylindrical shell

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. The aim of the present work is to investigate the influence of several modal geometrical imperfections on the nonlinear vibration of simply supported transversally excited cylindrical shells 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. Several numerical strategies are used to study the nonlinear behavior of the imperfect shell. Special attention is given to the shape and the magnitude of the initial geometric imperfection on the resonance curves and bifurcations of simply supported transversally excited cylindrical shells with 1:1:1:1 internal resonance (four interacting modes).