Integrity Factors of the Basins of Attraction of an Axially Excited Cylindrical Shell

Abstract

The aim of this investigation is to analyze the integrity factors of the basins of attraction of cylindrical shells subjected to an axial load composed of a static pre-load plus a harmonic excitation. The shell is described by the Donnell nonlinear shallow shell theory. The lateral displacement field is approximated qualitatively by a modal expansion with two degrees of freedom, containing the basic vibration mode plus an axi-symmetric mode with twice the number of waves as the basic mode, thus describing consistently the asymmetry of the nonlinear displacement field of the cylindrical shell surface. The discretized nonlinear differential equations of motion are obtained by the Galerkin method. In this paper special attention is paid to the influence of static pre-loading on the integrity of the basins of attraction of the permanent response of the harmonically excited cylindrical shell. For this, we obtain the parametric instability and permanent escape boundaries in the force control space, the bifurcation diagrams in both the pre- and post-buckling wells and the evolution of the basins of attraction for increasing levels of static pre-load. From this parametric analysis the variation of the integrity factor with the applied load is analyzed. It is shown that the variation of the integrity factor of each basin of attraction provides the designer with essential information on the safety of each stable long term solution of the excited shell