Nonlinear free vibration analysis of doubly curved shells

In this work, Sanders–Koiter’s nonlinear shell theory is applied to study the nonlinear moderate-amplitude vibrations of doubly curved shells using two different approximations of the strain–displacement relations for shallow and non-shallow shells. The nonlinear equations of motion are determined by Lagrange equations. The displacement fields are approximated using an expansion of trigonometric functions that satisfy geometric (essential) and nonlinear natural boundary conditions. Therefore, the backbone curves are determined using multiple shooting method and an Euler–Newtonian predictor–corrector continuation algorithm; the Floquet theory is applied to determine the stability of the periodic solutions. The obtained backbone curves show multiple internal resonances due to the coupling between normal modes. The mode influence of some selected points on the backbone curves is depicted to analyze the internal resonances, which can represent loss of stability and sudden changes in the dynamic behavior of shells undergoing moderate-amplitude vibrations. Saddle–node, Newmark–Sacker and period-doubling bifurcations are observed.