Non-linear free periodic vibrations of open cylindrical shallow shells

This paper is concerned with the non-linear free periodic vibrations of thin, open, cylindrical and shallow shells vibrating in the geometrically non-linear regime. A multi-degree-of-freedom model with hierarchical basis functions is adopted and the principle of the virtual work is used to define the time domain equations of motion. These equations are transformed into the frequency domain by the harmonic balance method and are finally solved by an arc-length continuation method. Shells of different thicknesses and of different curvature radius are analysed, and the variation of the non-linear natural frequencies of these shells with the vibration amplitude are investigated in some detail. The variation of the mode shapes with the vibration amplitude is demonstrated. It is found that both softening and hardening spring effects occur and that the number of couplings between vibration modes is rather large in undamped shells.