February 19, 2024
Synthesis of geometrically nonlinear reduced-order models for distributed elastic systems based on the finite element method.
The report is devoted to the problems of development and verification of computational algorithms for the synthesis of compact models of the dynamics of continual elastic systems in a geometrically nonlinear formulation (primarily thin-walled structures: strings, membranes, beams, plates, shells) based on the finite element method. The approaches under consideration are based on the idea ofidentifying a nonlinear (quadratic-cubic) stiffness characteristic of an elastic system in its modal coordinates with the subsequent application of the apparatus of the theory of nonlinear normal modes and Poincaré normal forms to construct an invariant manifold tangent to the modal subspace of interest. The resulting dynamic reduced-order model takes into account the nonlinear elastic coupling of the working vibration modes with high-frequency longitudinal and volume modes of the structure, which ensures the correctness of the calculated nonlinear stiffness characteristic of the system for the selected principal coordinates. The developed algorithm is applied to a number of problems of nonlinear dynamics of strings and beams that admit an approximate analytical solution using asymptotic methods of nonlinear mechanics. The features of the software implementation of the presented method based on the ABAQUS finite element analysis software system are discussed.