Nonlinear forced vibration analysis of doubly curved shells via the parameterization method for invariant manifold

In this work, the nonlinear forced vibrations of doubly curved shells are studied. For this, the Forced Resonance Curves of four different shells were determined: a shallow cylindrical panel, a shallow spherical panel, a non-shallow spherical panel, and a hyperbolic paraboloid. To model the shells, the Koiter’s nonlinear shell theory, for both shallow and non-shallow shells, was applied. The forced resonance curves were determined using an adaptive harmonic balance method and through a reduced-order model (ROM) via parameterization method for invariant manifolds. The findings of this study reveal the complex dynamic behavior exhibited by doubly curved shells, with various types of bifurcations such as Saddle–Node, Neimark–Sacker, and Period Doubling bifurcations. Thanks to the general treatment of the forcing term implemented in the parameterization method, the results highlight how complex high-order resonances can be retrieved by the ROM, up to a comfortable range of vibration and forcing amplitudes tested. Finally, it clearly demonstrates how the Nonlinear Normal Modes as invariant manifolds provide accurate and efficient ROMs for nonlinear vibrations of shells.