A reduced-order model for the nonlinear dynamics of shallow shells

In this study, we derive a reduced-order model to describe the nonlinear dynamics of shallow shells. Starting from the Marguerre–von Kármán (MvK) theory, the discretization is obtained by restricting the set of admissible configurations through few Lagrangian parameters. The resulting reduced model features a polynomial elastic energy composed of a membrane contribution—formulated via a standard linear in-plane problem and yielding a quartic polynomial—and a bending contribution for isotropic materials, expressed through a quadratic polynomial. The associated Lagrange equations of motion are derived by considering the kinetic energy of both in-plane and transverse displacements while neglecting rotary inertia. This reduced model can be viewed as an extension of the formulation proposed in [1], and its application is oriented toward the analysis of the nonlinear dynamics of bistable composite shells [2,3], with the specific goal of constructing global dynamic stability maps. 1. Brunetti M., Vincenti A., Vidoli S., A class of morphing shell structures satisfying clamped boundary conditions, Int. J. Solid Struct. 82 (2016) 47–55. 2. Brunetti M., Mitura A., Romeo F., Warminski J., Nonlinear dynamics of bistable composite cantilever shells: An experimental and modelling study, Journal of Sound and Vibrations, https://doi.org/10.1016/j.jsv.2022.116779, 2022. 3. Mitura A., Brunetti M., Kloda L., Romeo F., Warminski J., Experimental nonlinear dynamic regimes for energy harvesting from cantilever bistable shells, Mechanical Systems and and Signal Processing, 206, https://doi.org/10.1016/j.ymssp.2023.110890, 2024.